# Complete analysis of maximally hyperentangled state via the weak cross-Kerr
nonlinearity

Zhi Zeng${}^{1,2,^{*}}$ 1Institute of Signal Processing and Transmission,
Nanjing University of Posts and Telecommunications, Nanjing, 210003, China
2Key Lab of Broadband Wireless Communication and Sensor Network Technology,
Ministry of Education, Nanjing University of Posts and Telecommunications,
Nanjing, 210003, China<EMAIL_ADDRESS>
###### Abstract

We present a simple method for the complete analysis of maximally
hyperentangled state in polarization and spatial-mode degrees of freedom
assisted by the weak cross-Kerr nonlinearity. Our method not only can be used
for two-photon hyperentangled Bell state analysis and three-photon
hyperentangled Greenberger-Horne-Zeilinger (GHZ) state analysis, but also is
suitable for $N$-photon hyperentangled GHZ state analysis. In our protocols,
the bit information of hyperentanglement is read out via the nonlinear
interaction, and the phase information is obtained by using linear optical
element and single photon detector. This approach is achievable with the
current technology, and will be useful for the practical high-capacity quantum
communication schemes.

##  1 Introduction

Quantum entanglement is the key resource in quantum computation and quantum
communication, and it has been widely used for quantum key distribution [1,
2], quantum teleportation [3], quantum dense coding [4], quantum entanglement
swapping [5], quantum secret sharing [6, 7, 8] and quantum secure direct
communication [9, 10, 11, 12]. Quantum simulation is very useful to the
quantum science and technology. Recently, the efficient simulation of
photosynthetic light harvesting has attracted much attention [13, 14, 15]. In
many quantum information processing tasks, the complete analysis of a set of
orthogonal entangled states is an indispensable step to read out information.
Among the all types of entangled state analysis, the Bell state analysis (BSA)
for two-qubit system and Greenberger-Horne-Zeilinger (GHZ) state analysis
(GSA) for multi-qubit system are mostly used. Research has showed that the
linear-optics-based BSA and GSA are impossible, and the optimal success
probability of them are 50% and 25%, respectively [16, 17]. An efficient way
to deal with this problem is using hyperentanglement [18], which is defined as
entanglement in more than one degree of freedom (DOF). In 1998, Kwiat _et al._
showed that the four polarization Bell states can be reliably identified just
using linear optical elements, resorting to an additional DOF (time-bin or
spatial-mode) [19]. In 2003, Walborn _et al._ presented a simplified method
for BSA assisted by auxiliary entanglement [20]. Subsequently, several other
DOFs such as linear momentum DOF [21], time-bin DOF [22] and orbital angular
momentum DOF [23] were used to accomplish polarization BSA in experiment. In
2013, Song _et al._ proposed a scheme for $N$-photon GSA using
hyperentanglement [24]. In 2014, frequency entanglement was utilized for the
complete polarization BSA and GSA [25]. In addition to BSA and GSA,
hyperentanglement also has many other important applications in quantum
information processing, such as deterministic entanglement purification
protocols [26, 27, 28, 29, 30], hyperentanglement purification and
concentration [31, 32, 33, 34, 35], hyper-parallel optical quantum computation
[36, 37], and measurement-device-independent quantum communication protocols
[38, 39].

Hyperentangled Bell state analysis (HBSA) and hyperentangled GHZ state
analysis (HGSA) are essential to many high-capacity quantum communication
schemes. However, it has been shown that the 16 hyperentangled Bell states in
two DOFs only can be classified into 7 groups via linear optics, and thus the
complete HBSA is not achieved [40, 41]. If the quantum nonlinear interaction
is introduced, complete HBSA and HGSA can be both realized [42, 43, 44, 45,
46, 47, 48, 49, 50, 51, 52]. In 2010, Sheng _et al._ proposed the first
complete HBSA scheme for polarization and spatial-mode hyperentanglement based
on the cross-Kerr nonlinearity [42]. In 2012, Xia _et al._ presented an
efficient HGSA scheme using a similar principle [43]. In 2016, Li _et al._
proposed the self-assisted complete hyperentangled state analysis scheme for
HBSA and HGSA, in which the discrimination process can be greatly simplified
[44]. In the past decade, the quantum-dot spin in optical microcavity and
nitrogen-vacancy center in resonator have also been used for hyperentangled
state analysis [47, 48, 49, 50, 51, 52]. An alternative method for complete
HBSA and HGSA is using the auxiliary entanglement in other DOFs of photon [53,
54, 55]. In 2017, Li _et al._ showed that the 16 hyperentangled Bell states
can be separated into 12 groups with the help of an auxiliary known entangled
state in the third DOF [53]. In 2019, Wang _et al._ proposed an efficient HBSA
scheme for polarization and the first longitudinal momentum hyperentanglement,
by using a fixed hyperentangled Bell state in frequency and the second
longitudinal momentum DOFs [54].

In this paper, we present an efficient and simple method for the complete
analysis of hyperentangled Bell state and hyperentangled GHZ state in
polarization and spatial-mode DOFs via the weak cross-Kerr nonlinearity. The
bit information of hyperentangled state are obtained through the measurement
on coherent probe state, and the relative phase information are determined by
using linear optical element and single photon detector. The principle of our
schemes is quite different from the self-assisted mechanism proposed by Li _et
al._ in 2016 [44], in which the cross-Kerr nonlinearity is also utilized. Our
method can be directly extended to the complete $N$-photon HGSA, and will have
useful applications in high-capacity long-distance quantum communication.

##  2 Complete analysis of hyperentangled Bell state

In this section, we first introduce the photon number quantum nondemolition
detector (QND), which is constructed by the weak cross-Kerr nonlinearity.
Then, we will detailly demonstrate our complete HBSA scheme for the
hyperentangled photons in polarization and spatial-mode DOFs.

###  2.1 The principle of photon number QND

The cross-Kerr effect is an interaction between a signal state
$|\psi\rangle_{s}$ and a probe coherent state $|\alpha\rangle_{p}$ in a
nonlinear medium with the Hamiltonian [56]

$\displaystyle H=\hbar\chi a^{\dagger}_{s}a_{s}a^{\dagger}_{p}a_{p}.$ (1)

$a^{\dagger}_{s}$ ($a^{\dagger}_{p}$) and $a_{s}$ ($a_{p}$) are the creation
and destruction operators of the signal (probe) state, respectively. $\chi$ is
the coupling strength of the nonlinearity, which depends on the property of
nonlinear material. Supposing the signal state is a superposition of the Fock
state as

$\displaystyle|\psi\rangle_{s}=a|0\rangle_{s}+b|1\rangle_{s}+c|2\rangle_{s},$
(2)

the effect of the cross-Kerr nonlinearity can be described as

$\displaystyle
U|\psi\rangle_{s}|\alpha\rangle_{p}=a|0\rangle_{s}|\alpha\rangle_{p}+b|1\rangle_{s}|\alpha
e^{i\theta}\rangle_{p}+c|2\rangle_{s}|\alpha e^{2i\theta}\rangle_{p}.$ (3)

Here, $\theta=\chi t$ and $t$ is the interaction time. The coherent probe
state will pick up a phase shift, which is proportional to the number of
photon in the signal state. Using the $X$ quadrature measurement, the
information of phase shift can be obtained, and then the number of photon in
the signal state can be measured without destroying the photons.

###  2.2 Complete HBSA for polarization-spatial-mode hyperentanglement

The two-photon hyperentangled Bell state in polarization and spatial-mode DOFs
can be written as

$\displaystyle|\Psi\rangle_{AB}=|\zeta\rangle_{P}\otimes|\eta\rangle_{S}.$ (4)

Here $A$ and $B$ represent the two entangled photons, and the subscripts $P$
and $S$ denote the polarization and spatial-mode DOFs, respectively.
$|\zeta\rangle_{P}$ can be one of the four polarization Bell states,

$\displaystyle|\phi^{\pm}\rangle_{P}=\frac{1}{\sqrt{2}}(|HH\rangle\pm|VV\rangle)_{AB},$
$\displaystyle|\psi^{\pm}\rangle_{P}=\frac{1}{\sqrt{2}}(|HV\rangle\pm|VH\rangle)_{AB}.$
(5)

$H$ and $V$ are the horizontal and vertical polarization states, respectively.
$|\eta\rangle_{S}$ can be one of the four spatial-mode Bell states,

$\displaystyle|\phi^{\pm}\rangle_{S}=\frac{1}{\sqrt{2}}(|a_{1}b_{1}\rangle\pm|a_{2}b_{2}\rangle)_{AB},$
$\displaystyle|\psi^{\pm}\rangle_{S}=\frac{1}{\sqrt{2}}(|a_{1}b_{2}\rangle\pm|a_{2}b_{1}\rangle)_{AB}.$
(6)

Here $a_{1}$ ($b_{1}$) and $a_{2}$ ($b_{2}$) are the two different possible
spatial-modes of photon $A$ ($B$). The 16 hyperentangled states contain two
kinds of information, namely the bit information and the relative phase
information. In our proposal, the bit information is identified through the
two QNDs, which are accessible with the current technology. The relative phase
information is read out assisted by the linear optical element and single
photon detector. Thus, the 16 hyperentangeld Bell states in two DOFs can be
completely distinguished.

Figure 1: Schematic diagram of our complete HBSA scheme for polarization and
spatial-mode hyperentanglement. The half-wave plates (HWPs) implement the bit-
flip operation [$|H\rangle\leftrightarrow|V\rangle$] on the polarization state
of photon. The polarization beam splitters (PBSs) transmit the horizontal
polarized photon with its path switched, and reflect the vertical one with the
path unchanged. The cross-Kerr nonlinear interactions generate the phase shift
of $\pm\theta$ on the coherent states $|\alpha\rangle$ and $|\beta\rangle$ if
photons appear in the corresponding spatial-modes. The beam splitters (BSs)
accomplish the Hadamard operation
[$|a_{1}\rangle\rightarrow\frac{1}{\sqrt{2}}(|a_{1}\rangle+|a_{2}\rangle),|a_{2}\rangle\rightarrow\frac{1}{\sqrt{2}}(|a_{1}\rangle-|a_{2}\rangle),|b_{1}\rangle\rightarrow\frac{1}{\sqrt{2}}(|b_{1}\rangle+|b_{2}\rangle),|b_{2}\rangle\rightarrow\frac{1}{\sqrt{2}}(|b_{1}\rangle-|b_{2}\rangle)$]
on the spatial-mode state of photon. The wave plates (WPs) perform the
Hadamard operation
[$|H\rangle\rightarrow\frac{1}{\sqrt{2}}(|H\rangle+|V\rangle),|V\rangle\rightarrow\frac{1}{\sqrt{2}}(|H\rangle-|V\rangle)$]
on the polarization state of photon. Single photon detectors are required to
realize the polarization measurement in $\\{|H\rangle,|V\rangle\\}$ basis.
With this device, the 16 hyperentangled Bell states can be completely
distinguished.

The setup of our HBSA scheme for polarization-spatial-mode hyperentanglement
is shown in Fig. 1. After the two photons pass through HWPs and PBSs, interact
with the coherent state $|\alpha\rangle$, pass through the PBSs and HWPs
again, then the quantum state of the collective system composed of
hyperentanglement and coherent state evolves as

$\displaystyle|\phi^{\pm}\rangle_{P}|\eta\rangle_{S}|\alpha\rangle$
$\displaystyle\rightarrow$
$\displaystyle|\phi^{\pm}\rangle_{P}|\eta\rangle_{S}|\alpha\rangle,$
$\displaystyle|\psi^{\pm}\rangle_{P}|\eta\rangle_{S}|\alpha\rangle$
$\displaystyle\rightarrow$
$\displaystyle|\psi^{\pm}\rangle_{P}|\eta\rangle_{S}|\alpha e^{\pm
i\theta}\rangle.$ (7)

We can find that the entanglement in spatial-mode DOF is invariant during the
evolution. After the $X$-quadrature measurement on $|\alpha\rangle$, the bit
information of polarization entangled state is obtained. Subsequently, the
photons $A$ and $B$ will interact with coherent state $|\beta\rangle$, and the
evolution of the collective system is

$\displaystyle|\zeta\rangle_{P}|\phi^{\pm}\rangle_{S}|\beta\rangle$
$\displaystyle\rightarrow$
$\displaystyle|\zeta\rangle_{P}|\phi^{\pm}\rangle_{S}|\beta\rangle,$
$\displaystyle|\zeta\rangle_{P}|\psi^{\pm}\rangle_{S}|\beta\rangle$
$\displaystyle\rightarrow$
$\displaystyle|\zeta\rangle_{P}|\psi^{\pm}\rangle_{S}|\beta e^{\pm
i\theta}\rangle.$ (8)

It is shown that the polarization entanglement is invariant in the
interaction, and we can determine the bit information of spatial-mode
entanglement by utilizing the $X$-quadrature measurement on $|\beta\rangle$.
Therefore, the 16 hyperentangled Bell states can be classified into four
groups with the help of the coherent states $|\alpha\rangle$ and
$|\beta\rangle$, as shown in Table 1.

Table 1: Corresponding relations between the initial states and phase shifts of the two coherent states. Initial states |  |  |  | $|\alpha\rangle$ |  |  |  | $|\beta\rangle$
---|---|---|---|---|---|---|---|---
$|\phi^{+}\rangle_{P}|\phi^{\pm}\rangle_{S}$, $|\phi^{-}\rangle_{P}|\phi^{\pm}\rangle_{S}$. |  |  |  | $0$ |  |  |  | $0$
$|\phi^{+}\rangle_{P}|\psi^{\pm}\rangle_{S}$, $|\phi^{-}\rangle_{P}|\psi^{\pm}\rangle_{S}$. |  |  |  | $0$ |  |  |  | $\pm\theta$
$|\psi^{+}\rangle_{P}|\phi^{\pm}\rangle_{S}$, $|\psi^{-}\rangle_{P}|\phi^{\pm}\rangle_{S}$. |  |  |  | $\pm\theta$ |  |  |  | $0$
$|\psi^{+}\rangle_{P}|\psi^{\pm}\rangle_{S}$, $|\psi^{-}\rangle_{P}|\psi^{\pm}\rangle_{S}$. |  |  |  | $\pm\theta$ |  |  |  | $\pm\theta$

In each group, there are four states that have the same parity information but
different relative phase information, and these four states can be
distinguished by using the BSs and WPs. Here, we take the second group in
Table 1 as an example to illustrate. After passing through BSs and WPs, the
evolution of the hyperentangled state is

$\displaystyle|\phi^{+}\rangle_{P}|\psi^{+}\rangle_{S}\rightarrow\frac{1}{2}(|HH\rangle+|VV\rangle)\otimes(|a_{1}b_{1}\rangle-|a_{2}b_{2}\rangle)_{AB},$
$\displaystyle|\phi^{+}\rangle_{P}|\psi^{-}\rangle_{S}\rightarrow\frac{1}{2}(|HH\rangle+|VV\rangle)\otimes(|a_{1}b_{2}\rangle-|a_{2}b_{1}\rangle)_{AB},$
$\displaystyle|\phi^{-}\rangle_{P}|\psi^{+}\rangle_{S}\rightarrow\frac{1}{2}(|HV\rangle+|VH\rangle)\otimes(|a_{1}b_{1}\rangle-|a_{2}b_{2}\rangle)_{AB},$
$\displaystyle|\phi^{-}\rangle_{P}|\psi^{-}\rangle_{S}\rightarrow\frac{1}{2}(|HV\rangle+|VH\rangle)\otimes(|a_{1}b_{2}\rangle-|a_{2}b_{1}\rangle)_{AB}.$
(9)

According to the detection responses of single photon detectors, the above
four states can be determinately identified. In Table 2, we list the possible
initial hyperentangled Bell states with their corresponding detection results.
Using Table 1 and Table 2 simultaneously, we can unambiguously distinguish the
16 hyperentangled Bell states in polarization and spatial-mode DOFs.

Table 2: Corresponding relations between the initial 16 hyperentangled Bell states and the possible detection results of single photon detectors. Initial states |  |  |  | Possible detections
---|---|---|---|---
$|\phi^{+}\rangle_{P}|\phi^{+}\rangle_{S}$,$|\phi^{+}\rangle_{P}|\psi^{+}\rangle_{S}$,$|\psi^{+}\rangle_{P}|\phi^{+}\rangle_{S}$,$|\psi^{+}\rangle_{P}|\psi^{+}\rangle_{S}$. |  |  |  |  $A^{+}_{1}B^{+}_{1}$, $A^{-}_{1}B^{-}_{1}$, $A^{+}_{2}B^{+}_{2}$, $A^{-}_{2}B^{-}_{2}$.
$|\phi^{+}\rangle_{P}|\phi^{-}\rangle_{S}$,$|\phi^{+}\rangle_{P}|\psi^{-}\rangle_{S}$,$|\psi^{+}\rangle_{P}|\phi^{-}\rangle_{S}$,$|\psi^{+}\rangle_{P}|\psi^{-}\rangle_{S}$. |  |  |  |  $A^{+}_{1}B^{+}_{2}$, $A^{-}_{1}B^{-}_{2}$, $A^{+}_{2}B^{+}_{1}$, $A^{-}_{2}B^{-}_{1}$.
$|\phi^{-}\rangle_{P}|\phi^{+}\rangle_{S}$,$|\phi^{-}\rangle_{P}|\psi^{+}\rangle_{S}$,$|\psi^{-}\rangle_{P}|\phi^{+}\rangle_{S}$,$|\psi^{-}\rangle_{P}|\psi^{+}\rangle_{S}$. |  |  |  |  $A^{+}_{1}B^{-}_{1}$, $A^{-}_{1}B^{+}_{1}$, $A^{+}_{2}B^{-}_{2}$, $A^{-}_{2}B^{+}_{2}$.
$|\phi^{-}\rangle_{P}|\phi^{-}\rangle_{S}$,$|\phi^{-}\rangle_{P}|\psi^{-}\rangle_{S}$,$|\psi^{-}\rangle_{P}|\phi^{-}\rangle_{S}$,$|\psi^{-}\rangle_{P}|\psi^{-}\rangle_{S}$. |  |  |  |  $A^{+}_{1}B^{-}_{2}$, $A^{-}_{1}B^{+}_{2}$, $A^{+}_{2}B^{-}_{1}$, $A^{-}_{2}B^{+}_{1}$.

##  3 Complete analysis of hyperentangled Greenberger-Horne-Zeilinger state

The $N$-photon hyperentangled GHZ state in polarization and spatial-mode DOFs
can be written as

$\displaystyle|\Phi\rangle_{AB\cdots
N}=|\Phi\rangle_{P}\otimes|\Phi\rangle_{S},$ (10)

in which $A,B\cdots N$ represent the $N$ entangled photons. $|\Phi\rangle_{P}$
($|\Phi\rangle_{S}$) can be one of the $2^{N}$ polarization (spatial-mode) GHZ
states,

$\displaystyle|\Phi^{\pm}_{ij\cdots
k}\rangle_{P(S)}=\frac{1}{\sqrt{2}}(|ij\cdots
k\rangle\pm|\bar{i}\bar{j}\cdots\bar{k}\rangle)_{AB\cdots N}.$ (11)

Here, $i,j\cdots k\in\\{0,1\\}$ and $m=1-\bar{m}(m=i,j\cdots k)$. For the
polarization DOF, $|0\rangle\equiv|H\rangle$ and $|1\rangle\equiv|V\rangle$.
For the spatial-mode DOF, $|0\rangle\equiv|x_{1}\rangle$ and
$|1\rangle\equiv|x_{2}\rangle(x=a,b\cdots n)$. $|x_{1}\rangle$ and
$|x_{2}\rangle$ are the two possible spatial-modes of photon $X(X=A,B\cdots
N)$. Taking into account polarization and spatial-mode DOFs together, there
are $4^{N}$ hyperentangled GHZ states for the $N$-photon system, which will be
distinguished in the following text. We first describe the complete HGSA
scheme for three-photon system, and then generalize the scheme to $N$-photon
HGSA directly.

###  3.1 Complete HGSA for the three-photon hyperentanglement

When $N=3$, one of the 64 hyperentangled GHZ states in two DOFs can be written
as

$\displaystyle|\Phi^{+}_{000}\rangle_{P}\otimes|\Phi^{+}_{000}\rangle_{S}=\frac{1}{\sqrt{2}}(|HHH\rangle+|VVV\rangle)\otimes\frac{1}{\sqrt{2}}(|a_{1}b_{1}c_{1}\rangle+|a_{2}b_{2}c_{2}\rangle)_{ABC}.$

Figure 2: Schematic diagram of our complete HGSA scheme for the three-photon
hyperentanglement. Four coherent states are utilized to determine the parity
information of hyperentangled GHZ states, then the relative phase information
can be obtained by using linear optical elements and single photon detectors.
With this device, the 64 hyperentangled GHZ states in polarization and
spatial-mode DOFs can be completely distinguished.

In our three-photon HGSA scheme, four QNDs are required to get the parity
information of hyperentanglement, as shown in Fig. 2. After photons $A$, $B$
and $C$ pass through HWPs and PBSs, interact with the coherent states
$|\alpha_{1}\rangle$ and $|\alpha_{2}\rangle$, pass through the PBSs and HWPs
again, the quantum state of the collective system evolves as

$\displaystyle|\Phi^{\pm}_{000}\rangle_{P}|\Phi\rangle_{S}|\alpha_{1}\rangle|\alpha_{2}\rangle$
$\displaystyle\rightarrow$
$\displaystyle|\Phi^{\pm}_{000}\rangle_{P}|\Phi\rangle_{S}|\alpha_{1}\rangle|\alpha_{2}\rangle,$
$\displaystyle|\Phi^{\pm}_{001}\rangle_{P}|\Phi\rangle_{S}|\alpha_{1}\rangle|\alpha_{2}\rangle$
$\displaystyle\rightarrow$
$\displaystyle|\Phi^{\pm}_{001}\rangle_{P}|\Phi\rangle_{S}|\alpha_{1}\rangle|\alpha_{2}e^{\pm
i\theta}\rangle,$
$\displaystyle|\Phi^{\pm}_{010}\rangle_{P}|\Phi\rangle_{S}|\alpha_{1}\rangle|\alpha_{2}\rangle$
$\displaystyle\rightarrow$
$\displaystyle|\Phi^{\pm}_{010}\rangle_{P}|\Phi\rangle_{S}|\alpha_{1}e^{\pm
i\theta}\rangle|\alpha_{2}\rangle,$
$\displaystyle|\Phi^{\pm}_{100}\rangle_{P}|\Phi\rangle_{S}|\alpha_{1}\rangle|\alpha_{2}\rangle$
$\displaystyle\rightarrow$
$\displaystyle|\Phi^{\pm}_{100}\rangle_{P}|\Phi\rangle_{S}|\alpha_{1}e^{\pm
i\theta}\rangle|\alpha_{2}e^{\pm i\theta}\rangle.$ (13)

The spatial-mode entanglement is invariant during the evolution, and the
parity information of polarization entanglement can be obtained assisted by
the measurement on the two coherent states. After the three photons interact
with the coherent states $|\beta_{1}\rangle$ and $|\beta_{2}\rangle$, the
quantum state of the collective system evolves as

$\displaystyle|\Phi\rangle_{P}|\Phi^{\pm}_{000}\rangle_{S}|\beta_{1}\rangle|\beta_{2}\rangle$
$\displaystyle\rightarrow$
$\displaystyle|\Phi\rangle_{P}|\Phi^{\pm}_{000}\rangle_{S}|\beta_{1}\rangle|\beta_{2}\rangle,$
$\displaystyle|\Phi\rangle_{P}|\Phi^{\pm}_{001}\rangle_{S}|\beta_{1}\rangle|\beta_{2}\rangle$
$\displaystyle\rightarrow$
$\displaystyle|\Phi\rangle_{P}|\Phi^{\pm}_{001}\rangle_{S}|\beta_{1}\rangle|\beta_{2}e^{\pm
i\theta}\rangle,$
$\displaystyle|\Phi\rangle_{P}|\Phi^{\pm}_{010}\rangle_{S}|\beta_{1}\rangle|\beta_{2}\rangle$
$\displaystyle\rightarrow$
$\displaystyle|\Phi\rangle_{P}|\Phi^{\pm}_{010}\rangle_{S}|\beta_{1}e^{\pm
i\theta}\rangle|\beta_{2}\rangle,$
$\displaystyle|\Phi\rangle_{P}|\Phi^{\pm}_{100}\rangle_{S}|\beta_{1}\rangle|\beta_{2}\rangle$
$\displaystyle\rightarrow$
$\displaystyle|\Phi\rangle_{P}|\Phi^{\pm}_{100}\rangle_{S}|\beta_{1}e^{\pm
i\theta}\rangle|\beta_{2}e^{\pm i\theta}\rangle.$ (14)

It is easy to find that the polarization entanglement is unchanged, and the
parity information of spatial-mode entanglement is accessible. With the help
of the four QNDs, the 64 hyperentangled GHZ states thus can be divided into 16
groups, as shown in Table 3.

Table 3: Corresponding relations between the initial states and phase shifts of the four coherent states. Initial states |  |  |  | $|\alpha_{1}\rangle$ |  |  |  | $|\alpha_{2}\rangle$ |  |  |  | $|\beta_{1}\rangle$ |  |  |  | $|\beta_{2}\rangle$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$|\Phi^{+}_{000}\rangle_{P}|\Phi^{\pm}_{000}\rangle_{S}$, $|\Phi^{-}_{000}\rangle_{P}|\Phi^{\pm}_{000}\rangle_{S}$. |  |  |  | $0$ |  |  |  | $0$ |  |  |  | $0$ |  |  |  | $0$
$|\Phi^{+}_{000}\rangle_{P}|\Phi^{\pm}_{001}\rangle_{S}$, $|\Phi^{-}_{000}\rangle_{P}|\Phi^{\pm}_{001}\rangle_{S}$. |  |  |  | $0$ |  |  |  | $0$ |  |  |  | $0$ |  |  |  | $\pm\theta$
$|\Phi^{+}_{000}\rangle_{P}|\Phi^{\pm}_{010}\rangle_{S}$, $|\Phi^{-}_{000}\rangle_{P}|\Phi^{\pm}_{010}\rangle_{S}$. |  |  |  | $0$ |  |  |  | $0$ |  |  |  | $\pm\theta$ |  |  |  | $0$
$|\Phi^{+}_{000}\rangle_{P}|\Phi^{\pm}_{100}\rangle_{S}$, $|\Phi^{-}_{000}\rangle_{P}|\Phi^{\pm}_{100}\rangle_{S}$. |  |  |  | $0$ |  |  |  | $0$ |  |  |  | $\pm\theta$ |  |  |  | $\pm\theta$
$|\Phi^{+}_{001}\rangle_{P}|\Phi^{\pm}_{000}\rangle_{S}$, $|\Phi^{-}_{001}\rangle_{P}|\Phi^{\pm}_{000}\rangle_{S}$. |  |  |  | $0$ |  |  |  | $\pm\theta$ |  |  |  | $0$ |  |  |  | $0$
$|\Phi^{+}_{001}\rangle_{P}|\Phi^{\pm}_{001}\rangle_{S}$, $|\Phi^{-}_{001}\rangle_{P}|\Phi^{\pm}_{001}\rangle_{S}$. |  |  |  | $0$ |  |  |  | $\pm\theta$ |  |  |  | $0$ |  |  |  | $\pm\theta$
$|\Phi^{+}_{001}\rangle_{P}|\Phi^{\pm}_{010}\rangle_{S}$, $|\Phi^{-}_{001}\rangle_{P}|\Phi^{\pm}_{010}\rangle_{S}$. |  |  |  | $0$ |  |  |  | $\pm\theta$ |  |  |  | $\pm\theta$ |  |  |  | $0$
$|\Phi^{+}_{001}\rangle_{P}|\Phi^{\pm}_{100}\rangle_{S}$, $|\Phi^{-}_{001}\rangle_{P}|\Phi^{\pm}_{100}\rangle_{S}$. |  |  |  | $0$ |  |  |  | $\pm\theta$ |  |  |  | $\pm\theta$ |  |  |  | $\pm\theta$
$|\Phi^{+}_{010}\rangle_{P}|\Phi^{\pm}_{000}\rangle_{S}$, $|\Phi^{-}_{010}\rangle_{P}|\Phi^{\pm}_{000}\rangle_{S}$. |  |  |  | $\pm\theta$ |  |  |  | $0$ |  |  |  | $0$ |  |  |  | $0$
$|\Phi^{+}_{010}\rangle_{P}|\Phi^{\pm}_{001}\rangle_{S}$, $|\Phi^{-}_{010}\rangle_{P}|\Phi^{\pm}_{001}\rangle_{S}$. |  |  |  | $\pm\theta$ |  |  |  | $0$ |  |  |  | $0$ |  |  |  | $\pm\theta$
$|\Phi^{+}_{010}\rangle_{P}|\Phi^{\pm}_{010}\rangle_{S}$, $|\Phi^{-}_{010}\rangle_{P}|\Phi^{\pm}_{010}\rangle_{S}$. |  |  |  | $\pm\theta$ |  |  |  | $0$ |  |  |  | $\pm\theta$ |  |  |  | $0$
$|\Phi^{+}_{010}\rangle_{P}|\Phi^{\pm}_{100}\rangle_{S}$, $|\Phi^{-}_{010}\rangle_{P}|\Phi^{\pm}_{100}\rangle_{S}$. |  |  |  | $\pm\theta$ |  |  |  | $0$ |  |  |  | $\pm\theta$ |  |  |  | $\pm\theta$
$|\Phi^{+}_{100}\rangle_{P}|\Phi^{\pm}_{000}\rangle_{S}$, $|\Phi^{-}_{100}\rangle_{P}|\Phi^{\pm}_{000}\rangle_{S}$. |  |  |  | $\pm\theta$ |  |  |  | $\pm\theta$ |  |  |  | $0$ |  |  |  | $0$
$|\Phi^{+}_{100}\rangle_{P}|\Phi^{\pm}_{001}\rangle_{S}$, $|\Phi^{-}_{100}\rangle_{P}|\Phi^{\pm}_{001}\rangle_{S}$. |  |  |  | $\pm\theta$ |  |  |  | $\pm\theta$ |  |  |  | $0$ |  |  |  | $\pm\theta$
$|\Phi^{+}_{100}\rangle_{P}|\Phi^{\pm}_{010}\rangle_{S}$, $|\Phi^{-}_{100}\rangle_{P}|\Phi^{\pm}_{010}\rangle_{S}$. |  |  |  | $\pm\theta$ |  |  |  | $\pm\theta$ |  |  |  | $\pm\theta$ |  |  |  | $0$
$|\Phi^{+}_{100}\rangle_{P}|\Phi^{\pm}_{100}\rangle_{S}$, $|\Phi^{-}_{100}\rangle_{P}|\Phi^{\pm}_{100}\rangle_{S}$. |  |  |  | $\pm\theta$ |  |  |  | $\pm\theta$ |  |  |  | $\pm\theta$ |  |  |  | $\pm\theta$

In each group, there are four states with the same parity information but
different relative phase information, which will be distinguished in the next
step. We take the second group in Table 3 as an example to illustrate, and the
evolution of the hyperentanglement after passing through BSs and WPs will be

$\displaystyle|\Phi^{+}_{000}\rangle_{P}|\Phi^{+}_{001}\rangle_{S}\rightarrow$
$\displaystyle\frac{1}{4}(|HHH\rangle+|HVV\rangle+|VHV\rangle+|VVH\rangle)$
$\displaystyle\otimes(|a_{1}b_{1}c_{1}\rangle-|a_{1}b_{2}c_{2}\rangle-|a_{2}b_{1}c_{2}\rangle+|a_{2}b_{2}c_{1}\rangle)_{ABC},$
$\displaystyle|\Phi^{+}_{000}\rangle_{P}|\Phi^{-}_{001}\rangle_{S}\rightarrow$
$\displaystyle\frac{1}{4}(|HHH\rangle+|HVV\rangle+|VHV\rangle+|VVH\rangle)$
$\displaystyle\otimes(|a_{1}b_{1}c_{2}\rangle-|a_{1}b_{2}c_{1}\rangle-|a_{2}b_{1}c_{1}\rangle+|a_{2}b_{2}c_{2}\rangle)_{ABC},$
$\displaystyle|\Phi^{-}_{000}\rangle_{P}|\Phi^{+}_{001}\rangle_{S}\rightarrow$
$\displaystyle\frac{1}{4}(|HHV\rangle+|HVH\rangle+|VHH\rangle+|VVV\rangle)$
$\displaystyle\otimes(|a_{1}b_{1}c_{1}\rangle-|a_{1}b_{2}c_{2}\rangle-|a_{2}b_{1}c_{2}\rangle+|a_{2}b_{2}c_{1}\rangle)_{ABC},$
$\displaystyle|\Phi^{-}_{000}\rangle_{P}|\Phi^{-}_{001}\rangle_{S}\rightarrow$
$\displaystyle\frac{1}{4}(|HHV\rangle+|HVH\rangle+|VHH\rangle+|VVV\rangle)$
(15)
$\displaystyle\otimes(|a_{1}b_{1}c_{2}\rangle-|a_{1}b_{2}c_{1}\rangle-|a_{2}b_{1}c_{1}\rangle+|a_{2}b_{2}c_{2}\rangle)_{ABC}.$

These four states are distinguishable through the detection results of single
photon detectors, with which all the 64 hyperentangled GHZ states can be
classified into four groups, as shown in Table 4. There are 16 hyperentangled
states in each group, and we can perfectly identify them with the help of
Table 3. Therefore, the complete HGSA scheme for three-photon system is
achievable by using Table 3 and Table 4 simultaneously.

Table 4: The 64 hyperentangled GHZ states can be classified into four groups based on the detection results of single photon detectors. Group | Initial states
---|---
$1$ |  $|\Phi^{+}_{000}\rangle_{P}|\Phi^{+}_{000}\rangle_{S},|\Phi^{+}_{000}\rangle_{P}|\Phi^{+}_{001}\rangle_{S},$ $|\Phi^{+}_{000}\rangle_{P}|\Phi^{+}_{010}\rangle_{S},|\Phi^{+}_{000}\rangle_{P}|\Phi^{+}_{100}\rangle_{S},$
|
$|\Phi^{+}_{001}\rangle_{P}|\Phi^{+}_{000}\rangle_{S},|\Phi^{+}_{001}\rangle_{P}|\Phi^{+}_{001}\rangle_{S},$
$|\Phi^{+}_{001}\rangle_{P}|\Phi^{+}_{010}\rangle_{S},|\Phi^{+}_{001}\rangle_{P}|\Phi^{+}_{100}\rangle_{S},$
|
$|\Phi^{+}_{010}\rangle_{P}|\Phi^{+}_{000}\rangle_{S},|\Phi^{+}_{010}\rangle_{P}|\Phi^{+}_{001}\rangle_{S},$
$|\Phi^{+}_{010}\rangle_{P}|\Phi^{+}_{010}\rangle_{S},|\Phi^{+}_{010}\rangle_{P}|\Phi^{+}_{100}\rangle_{S},$
|
$|\Phi^{+}_{100}\rangle_{P}|\Phi^{+}_{000}\rangle_{S},|\Phi^{+}_{100}\rangle_{P}|\Phi^{+}_{001}\rangle_{S},$
$|\Phi^{+}_{100}\rangle_{P}|\Phi^{+}_{010}\rangle_{S},|\Phi^{+}_{100}\rangle_{P}|\Phi^{+}_{100}\rangle_{S}.$
$2$ |  $|\Phi^{+}_{000}\rangle_{P}|\Phi^{-}_{000}\rangle_{S},|\Phi^{+}_{000}\rangle_{P}|\Phi^{-}_{001}\rangle_{S},$ $|\Phi^{+}_{000}\rangle_{P}|\Phi^{-}_{010}\rangle_{S},|\Phi^{+}_{000}\rangle_{P}|\Phi^{-}_{100}\rangle_{S},$
|
$|\Phi^{+}_{001}\rangle_{P}|\Phi^{-}_{000}\rangle_{S},|\Phi^{+}_{001}\rangle_{P}|\Phi^{-}_{001}\rangle_{S},$
$|\Phi^{+}_{001}\rangle_{P}|\Phi^{-}_{010}\rangle_{S},|\Phi^{+}_{001}\rangle_{P}|\Phi^{-}_{100}\rangle_{S},$
|
$|\Phi^{+}_{010}\rangle_{P}|\Phi^{-}_{000}\rangle_{S},|\Phi^{+}_{010}\rangle_{P}|\Phi^{-}_{001}\rangle_{S},$
$|\Phi^{+}_{010}\rangle_{P}|\Phi^{-}_{010}\rangle_{S},|\Phi^{+}_{010}\rangle_{P}|\Phi^{-}_{100}\rangle_{S},$
|
$|\Phi^{+}_{100}\rangle_{P}|\Phi^{-}_{000}\rangle_{S},|\Phi^{+}_{100}\rangle_{P}|\Phi^{-}_{001}\rangle_{S},$
$|\Phi^{+}_{100}\rangle_{P}|\Phi^{-}_{010}\rangle_{S},|\Phi^{+}_{100}\rangle_{P}|\Phi^{-}_{100}\rangle_{S}.$
$3$ |  $|\Phi^{-}_{000}\rangle_{P}|\Phi^{+}_{000}\rangle_{S},|\Phi^{-}_{000}\rangle_{P}|\Phi^{+}_{001}\rangle_{S},$ $|\Phi^{-}_{000}\rangle_{P}|\Phi^{+}_{010}\rangle_{S},|\Phi^{-}_{000}\rangle_{P}|\Phi^{+}_{100}\rangle_{S},$
|
$|\Phi^{-}_{001}\rangle_{P}|\Phi^{+}_{000}\rangle_{S},|\Phi^{-}_{001}\rangle_{P}|\Phi^{+}_{001}\rangle_{S},$
$|\Phi^{-}_{001}\rangle_{P}|\Phi^{+}_{010}\rangle_{S},|\Phi^{-}_{001}\rangle_{P}|\Phi^{+}_{100}\rangle_{S},$
|
$|\Phi^{-}_{010}\rangle_{P}|\Phi^{+}_{000}\rangle_{S},|\Phi^{-}_{010}\rangle_{P}|\Phi^{+}_{001}\rangle_{S},$
$|\Phi^{-}_{010}\rangle_{P}|\Phi^{+}_{010}\rangle_{S},|\Phi^{-}_{010}\rangle_{P}|\Phi^{+}_{100}\rangle_{S},$
|
$|\Phi^{-}_{100}\rangle_{P}|\Phi^{+}_{000}\rangle_{S},|\Phi^{-}_{100}\rangle_{P}|\Phi^{+}_{001}\rangle_{S},$
$|\Phi^{-}_{100}\rangle_{P}|\Phi^{+}_{010}\rangle_{S},|\Phi^{-}_{100}\rangle_{P}|\Phi^{+}_{100}\rangle_{S}.$
$4$ |  $|\Phi^{-}_{000}\rangle_{P}|\Phi^{-}_{000}\rangle_{S},|\Phi^{-}_{000}\rangle_{P}|\Phi^{-}_{001}\rangle_{S},$ $|\Phi^{-}_{000}\rangle_{P}|\Phi^{-}_{010}\rangle_{S},|\Phi^{-}_{000}\rangle_{P}|\Phi^{-}_{100}\rangle_{S},$
|
$|\Phi^{-}_{001}\rangle_{P}|\Phi^{-}_{000}\rangle_{S},|\Phi^{-}_{001}\rangle_{P}|\Phi^{-}_{001}\rangle_{S},$
$|\Phi^{-}_{001}\rangle_{P}|\Phi^{-}_{010}\rangle_{S},|\Phi^{-}_{001}\rangle_{P}|\Phi^{-}_{100}\rangle_{S},$
|
$|\Phi^{-}_{010}\rangle_{P}|\Phi^{-}_{000}\rangle_{S},|\Phi^{-}_{010}\rangle_{P}|\Phi^{-}_{001}\rangle_{S},$
$|\Phi^{-}_{010}\rangle_{P}|\Phi^{-}_{010}\rangle_{S},|\Phi^{-}_{010}\rangle_{P}|\Phi^{-}_{100}\rangle_{S},$
|
$|\Phi^{-}_{100}\rangle_{P}|\Phi^{-}_{000}\rangle_{S},|\Phi^{-}_{100}\rangle_{P}|\Phi^{-}_{001}\rangle_{S},$
$|\Phi^{-}_{100}\rangle_{P}|\Phi^{-}_{010}\rangle_{S},|\Phi^{-}_{100}\rangle_{P}|\Phi^{-}_{100}\rangle_{S}.$

###  3.2 Complete HGSA for the $N$-photon hyperentanglement

Our method is also suitable for the complete $N$-photon HGSA, and the setup of
our protocol is shown in Fig. 3. Like the complete HBSA and three-photon HGSA,
the complete $N$-photon HGSA can be accomplished via three steps.

Figure 3: Schematic diagram of our complete HGSA scheme for the $N$-photon
hyperentanglement. With this device, the $4^{N}$ hyperentangled GHZ states in
two DOFs can be completely distinguished.

Firstly, the parity information of polarization entanglement is determined
without affecting the spatial-mode entanglement, resorting to the measurements
on $N-1$ coherent states. This process can be expressed as

$\displaystyle|\Phi^{\pm}_{ijp\cdots
qk}\rangle_{P}|\Phi\rangle_{S}|\alpha_{1}\rangle|\alpha_{2}\rangle\cdots|\alpha_{N-2}\rangle|\alpha_{N-1}\rangle$
$\displaystyle\rightarrow|\Phi^{\pm}_{ijp\cdots
qk}\rangle_{P}|\Phi\rangle_{S}|\alpha_{1}e^{\pm(i\oplus
j)i\theta}\rangle|\alpha_{2}e^{\pm(i\oplus
p)i\theta}\rangle\cdots|\alpha_{N-2}e^{\pm(i\oplus
q)i\theta}\rangle|\alpha_{N-1}e^{\pm(i\oplus k)i\theta}\rangle.$

Subsequently, the $N$ photons will interact with the other $N-1$ coherent
states, and thus the parity information of spatial-mode entanglement is
determined. This process can be expressed as

$\displaystyle|\Phi\rangle_{P}|\Phi^{\pm}_{ijp\cdots
qk}\rangle_{S}|\beta_{1}\rangle|\beta_{2}\rangle\cdots|\beta_{N-2}\rangle|\beta_{N-1}\rangle$
$\displaystyle\rightarrow|\Phi\rangle_{P}|\Phi^{\pm}_{ijp\cdots
qk}\rangle_{S}|\beta_{1}e^{\pm(i\oplus
j)i\theta}\rangle|\beta_{2}e^{\pm(i\oplus
p)i\theta}\rangle\cdots|\beta_{N-2}e^{\pm(i\oplus
q)i\theta}\rangle|\beta_{N-1}e^{\pm(i\oplus k)i\theta}\rangle.$

With these two steps, the $4^{N}$ hyperentangled GHZ states in two DOFs can be
separated into $x$ groups,

$\displaystyle x=2^{N-1}*2^{N-1}=4^{N-1}.$ (18)

Four states exist in each group, and they can be described as
$|\Phi^{+}_{ij\cdots k}\rangle_{P}|\Phi^{+}_{ij\cdots k}\rangle_{S}$,
$|\Phi^{+}_{ij\cdots k}\rangle_{P}|\Phi^{-}_{ij\cdots k}\rangle_{S}$,
$|\Phi^{-}_{ij\cdots k}\rangle_{P}|\Phi^{+}_{ij\cdots k}\rangle_{S}$ and
$|\Phi^{-}_{ij\cdots k}\rangle_{P}|\Phi^{-}_{ij\cdots k}\rangle_{S}$. Finally,
these four states will be distinguished with linear optics and single photon
detectors. Specifically speaking, the number of $|V\rangle$ and number of
$|x_{2}\rangle$ are both even in the detection results for
$|\Phi^{+}_{ij\cdots k}\rangle_{P}|\Phi^{+}_{ij\cdots k}\rangle_{S}$, since
the phase information of polarization state and spatial-mode state are both
”+”. For $|\Phi^{+}_{ij\cdots k}\rangle_{P}|\Phi^{-}_{ij\cdots k}\rangle_{S}$,
the detection results contain an even number of $|V\rangle$ and an odd number
of $|x_{2}\rangle$. For $|\Phi^{-}_{ij\cdots k}\rangle_{P}|\Phi^{+}_{ij\cdots
k}\rangle_{S}$, the detection results contain an odd number of $|V\rangle$ and
an even number of $|x_{2}\rangle$, while the detection results contain an odd
number of $|V\rangle$ and an odd number of $|x_{2}\rangle$ for
$|\Phi^{-}_{ij\cdots k}\rangle_{P}|\Phi^{-}_{ij\cdots k}\rangle_{S}$. Up to
now, the $4^{N}$ hyperentangled GHZ states have been completely distinguished.

##  4 Discussion and summary

In the presented schemes, we have proposed an efficient method for the
complete two-photon HBSA and multi-photon HGSA with the help of photon number
QND and linear optics. The whole discrimination process is accomplished via
three steps. Firstly, the parity information of polarization entanglement are
obtained by using $N-1$ QNDs for $N$-photon system. Then, other $N-1$ QNDs are
required for determining the parity information of spatial-mode entanglement.
At last, the relative phase information of hyperentanglement are identified by
using the linear optics. With these three steps, both of the complete HBSA and
HGSA can be realized. In essence, the quantum manipulation of photonic
hyperentangled state in two different DOFs are independent, that’s why our
schemes can operate successfully.

The cross-Kerr nonlinearity is used for constructing the two-photon QND, which
will mainly influence the feasibility of our schemes [56]. Although it has
been widely studied in the past years, the effective nonlinearity strength is
still challenging in the single-photon regime with the current technology [57,
58, 59]. Fortunately, we just need the weak cross-Kerr nonlinearity that could
produce the small phase shift in the coherent state. When a sufficiently large
amplitude of the coherent state satisfies $\alpha\theta^{2}\gg 1$ ($\theta$ is
the cross-phase shift), it is possible to distinguish the small phase of
coherent state from the zero phase shift. Actually, the theoretical and
experimental researches have shown that it is promising for us to construct
the QND required in our method, resorting to the weak cross-Kerr nonlinearity
[60, 61, 62, 63, 64, 65, 66, 67, 68]. In addition, some other kinds of
interaction can also provide accessible ways to realize the photon number QND,
such as cavity-assisted interactions [69] and quantum dot spin in optical
microcavity [70].

In summary, we have proposed an efficient method for the complete analysis of
maximally hyperentangled state in polarization and spatial-mode DOFs via the
weak cross-Kerr nonlinearity. This approach is achievable with the current
technology, and can be used for the complete HBSA and $N$-photon HGSA. Our
method provides an alternative way to the hyperentangled state analysis, and
will be useful for the high-capacity quantum communication based on
hyperentanglement.

## References

## References

  * [1] A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991).
  * [2] C. H. Bennett, G. Brassard, and N. D. Mermin, Phys. Rev. Lett. 68, 557 (1992).
  * [3] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).
  * [4] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992).
  * [5] M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, Phys. Rev. Lett. 71, 4287 (1993).
  * [6] M. Hillery, V. Bužek, and A. Berthiaume, Phys. Rev. A 59, 1829 (1999).
  * [7] A. Karlsson, M. Koashi, and N. Imoto, Phys. Rev. A 59, 162 (1999).
  * [8] L. Xiao, G. L. Long, F. G. Deng, and J. W. Pan, Phys. Rev. A 69, 052307 (2004).
  * [9] G. L. Long and X. S. Liu, Phys. Rev. A 65, 032302 (2002).
  * [10] F. G. Deng, G. L. Long, and X. S. Liu, Phys. Rev. A 68, 042317 (2003).
  * [11] F. G. Deng and G. L. Long, Phys. Rev. A 69, 052319 (2004).
  * [12] C. Wang, F. G. Deng, Y. S. Li, X. S. Liu, and G. L. Long, Phys. Rev. A 71, 044305 (2005).
  * [13] B. X. Wang, M. J. Tao, Q. Ai, T. Xin, N. Lambert, D. Ruan, Y. C. Cheng, F. Nori, F. G. Deng, and G. L. Long, npj Quantum Inf. 4, 52 (2018).
  * [14] M. J. Tao, M. Hua, N. N. Zhang, W. T. He, Q. Ai, and F. G. Deng, Quantum Eng. 2, e53 (2020).
  * [15] N. N. Zhang, M. J. Tao, W. T. He, X. Y. Chen, X. Y. Kong, F. G. Deng, N. Lambert, and Q. Ai, Front. Phys. 16, 51501 (2021).
  * [16] N. Lütkenhaus, J. Calsamiglia, and K. A. Suominen, Phys. Rev. A 59, 3295 (1999).
  * [17] J. W. Pan and A. Zeilinger, Phys. Rev. A 57, 2208 (1998).
  * [18] F. G. Deng, B. C. Ren, and X. H. Li, Sci. Bull. 62, 46 (2017).
  * [19] P. G. Kwiat and H. Weinfurter, Phys. Rev. A 58, R2623 (1998).
  * [20] S. P. Walborn, S. Pádua, and C. H. Monken, Phys. Rev. A 68, 042313 (2003).
  * [21] M. Barbieri, G. Vallone, P. Mataloni, and F. De Martini, Phys. Rev. A 75, 042317 (2007).
  * [22] C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter, Phys. Rev. Lett. 96, 190501 (2006).
  * [23] J. T. Barreiro, T. C. Wei, and P. G. Kwiat, Nat. Phys. 4, 282 (2008).
  * [24] S. Song, Y, Cao, Y. B. Sheng, G. L. Long, Quantum Inf. Process. 12, 381 (2013).
  * [25] Z. Zeng, C, Wang, X. H. Li, Commun. Theor. Phys. 62, 683 (2014).
  * [26] Y. B. Sheng and F. G. Deng, Phys. Rev. A 81, 032307 (2010).
  * [27] X. H. Li, Phys. Rev. A 82, 044304 (2010).
  * [28] Y. B. Sheng and F. G. Deng, Phys. Rev. A 82, 044305 (2010).
  * [29] F. G. Deng, Phys. Rev. A 83, 062316 (2011).
  * [30] Z. Zeng, C, Wang, X. H. Li, and H. Wei, Laser Phys. Lett. 12, 015201 (2015).
  * [31] B. C. Ren, F. F. Du, and F. G. Deng, Phys. Rev. A 90, 052309 (2014).
  * [32] G. Y. Wang, Q. Liu, and F. G. Deng, Phys. Rev. A 94, 032319 (2016).
  * [33] B. C. Ren, F. F. Du, and F. G. Deng, Phys. Rev. A 88, 012302 (2013).
  * [34] X. H. Li and S. Ghose, Opt. Express 23, 3550 (2015).
  * [35] X. H. Li and S. Ghose, Phys. Rev. A 91, 062302 (2015).
  * [36] T. Li and G. L. Long, Phys. Rev. A 94, 022343 (2016).
  * [37] G. Y. Wang, T. Li, Q. Ai, and F. G. Deng, Opt. Express 26, 23333 (2018).
  * [38] Y. F. Yan, L. Zhou, W. Zhong, and Y. B. Sheng, Front. Phys. 16, 11501 (2021).
  * [39] X. D. Wu, L. Zhou, W. Zhong, and Y. B. Sheng, Quant. Inf. Process. 19, 354 (2020).
  * [40] T. C. Wei, J. T. Barreiro, and P. G. Kwiat, Phys. Rev. A 75, 060305(R) (2007).
  * [41] N. Pisenti, C. P. E. Gaebler, and T. W. Lynn, Phys. Rev. A 84, 022340 (2011).
  * [42] Y. B. Sheng, F. G. Deng, and G. L. Long, Phys. Rev. A 82, 032318 (2010).
  * [43] Y. Xia, Q. Q. Chen, J. Song, and H. S. Song, J. Opt. Soc. Am. B 29, 1029 (2012).
  * [44] X. H. Li and S. Ghose, Phys. Rev. A 93, 022302 (2016).
  * [45] X. H. Li and S. Ghose, Opt. Express 24, 18388 (2016).
  * [46] Q. Liu, G. Y. Wang, Q. Ai, M. Zhang, and F. G. Deng, Sci. Reports 6, 22016 (2016).
  * [47] B. C. Ren, H. R. Wei, M. Hua, T. Li, and F. G. Deng, Opt. Express 20, 24664 (2012).
  * [48] Q. Liu, and M. Zhang, J. Opt. Soc. Am. B 30, 2263 (2013).
  * [49] Z. Zeng, Laser Phys. Lett. 15, 055204 (2018).
  * [50] C. Cao, L. Zhang, Y. H. Han, P. P. Yin, L. Fan, Y. W. Duan, and R. Zhang, Opt. Express 28, 2857 (2020).
  * [51] T. J. Wang, Y. Lu, and G. L. Long, Phys. Rev. A 86, 042337 (2012).
  * [52] Q. Liu and M. Zhang, Phys. Rev. A 91, 062321 (2015).
  * [53] X. H. Li and S. Ghose, Phys. Rev. A 96, 020303(R) (2017).
  * [54] G. Y. Wang, B. C. Ren, F. G. Deng and G. L. Long, Opt. Express 27, 8994 (2019).
  * [55] Z. Zeng and K. D. Zhu, New J. Phys. 22, 083051 (2020).
  * [56] K. Nemoto and W. J. Munro, Phys. Rev. Lett. 93, 250502 (2004).
  * [57] W. J. Munro, K. Nemoto and T. P. Spiller, New J. Phys. 7, 137 (2005).
  * [58] J. H. Shapiro, Phys. Rev. A 73, 062305 (2006).
  * [59] J. H. Shapiro and M. Razavi, New J. Phys. 9, 16 (2007).
  * [60] C. Wittmann, U. L. Andersen, M. Takeoka, D. Sych, and G. Leuchs, Phys. Rev. A 81, 062338 (2010).
  * [61] N. Matsuda, R. Shimizu, Y. Mitsumori, H. Kosaka, and K. Edamatsu, Nat. Photon. 3, 95 (2009).
  * [62] B. He, Q. Lin, and C. Simon, Phys. Rev. A 83, 053826 (2011).
  * [63] C. Zhu and G. Huang, Opt. Express 19, 23364 (2011).
  * [64] A. Feizpour, X. Xing, and A. M. Steinberg, Phys. Rev. Lett. 107, 133603 (2011).
  * [65] I. C. Hoi, A. F. Kockum, T. Palomaki, T.M. Stace, B. Fan, and L. Tornberg, Phys. Rev. Lett. 111, 053601 (2013).
  * [66] A. Feizpour, M. Hallaji, G. Dmochowski, and A. M. Steinberg, Nat. Phys. 11, 905 (2015).
  * [67] K. M. Beck, M. Hosseini, Y. H. Duan, and V. Vuletic, PNAS 113, 9740 (2016).
  * [68] D. Tiarks, S. Schmidt, G. Rempe, and S. Durr, Sci. Adv. 2, e1600036 (2016).
  * [69] L. M. Duan and H. J. Kimble, Phys. Rev. Lett. 92, 127902 (2004).
  * [70] C. Bonato, F. Haupt, S. S. R. Oemrawsingh, J. Gudat, D. P. Ding, M. P. van Exter, and D. Bouwmeester, Phys. Rev. Lett. 104, 160503 (2010).


Modified Erlang loss system for cognitive wireless networks E. V. Morozov, S.
S. Rogozin, H. Q. Nguyen, T. Phung-Duc E. V. Morozov, S. S. Rogozin, H. Q.
Nguyenand T. Phung-Duc Institute of Applied Mathematical Research, Karelian
Research Centre of RAS, Petrozavodsk, Russia;
e-mail<EMAIL_ADDRESS>(E. M<EMAIL_ADDRESS>(S. R.); Petrozavodsk State
University, Petrozavodsk, Russia; Moscow Center for Fundamental and Applied
Mathematics, Moscow State University, Moscow 119991, Russia; University of
Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan;
e-mail<EMAIL_ADDRESS>(H. Q. N<EMAIL_ADDRESS>(T. P.)
This paper considers a modified Erlang loss system for cognitive wireless
networks and related applications. A primary user has preemptive priority over
secondary users and the primary customer is lost if upon arrival all the
channels are used by other primary users. Secondary users cognitively use idle
channels and they can wait at an infinite buffer in cases idle channels are
not available upon arrival or they are interrupted by primary users. We obtain
explicit stability condition for the cases where arrival processes of primary
users and secondary users follow Poisson processes and their service times
follow two distinct arbitrary distributions. The stability condition is
insensitive to the service time distributions and implies the maximal
throughout of secondary users. For a special case of exponential service time
distributions, we analyze in depth to show the effect of parameters on the
delay performance and the mean number of interruptions of secondary users. Our
simulations for distributions rather than exponential reveal that the mean
number of terminations for secondary users is less sensitive to the service
time distribution of primary users.

##  1 Introduction

In recent years, Internet traffic has increased explosively due to the
increased use of smart-phones, tablet computers, etc. This causes a shortage
problem of wireless spectrum. Cognitive wireless is considered as a promising
solution to this problem [1, 4, 5, 2, 3]. For recent development of cognitive
radio networks, we refer to the survey paper by Ostovar et al. [6]. In
wireless networks, secondary users (unlicensed users) are allowed to
cognitively use the bandwidths that are originally allocated to primary users
(licensed users). Secondary users should use the bandwidths in such a way that
does not interfere primary users. In particular, secondary users can use the
bandwidths only if primary users are not present. To this end, secondary users
must be aware of the presence of primary users so as to evacuate upon arrivals
of primary users. From this point of view, primary users have absolute
priority over secondary users meaning that the transmission of a secondary
user might be interrupted by a primary user. We assume that interrupted
secondary users evacuate to the head of the buffer and resume their
transmission as soon as a channel is available.

Motivated by the above situation, we propose analyzing a multiserver queueing
system with an infinite buffer for secondary users while primary users have
absolute priority over secondary users and are lost if all channels are
already occupied by other primary users. Under this assumption (and under
Poisson inputs), from the view point of primary users, the system of servers
behaves as an Erlang loss system while from that of secondary users, the
system is an infinite buffer model where secondary users are served when some
servers are not occupied by primary users. In our model, the service times of
primary and secondary customers follow two distinct arbitrary distributions.

A closely related model is the paper by Mitrani and Avi-Itzhak [7]. In this
paper, the author considers an M/M/$N$ system where each server is subject to
random breakdowns and repairs. The author analyzes the joint distribution of
the queue length and the state of the servers using a generating function
approach. The model in [7] can be considered as a model with $N$ primary
customers. In our model, primary customers arrive according to a Poisson
process implying that there are infinite number of primary customers. Akutsu
and Phung-Duc [8] examine a closely related Markovian model in which secondary
customers first sense the channels before occupying them. The stability
condition is conjectured and verified by simulation in [8]. Salameh et al.
[10, 9] consider models with limited number of sensing secondary users. Other
queueing models of cognitive radio networks could be found in [16, 12, 13, 15,
17, 11, 14]. In [12, 13, 11, 18], the service time distributions of primary
and secondary customers are either restricted to Markovian distributions
(exponential or phase-type distributions) and/or the assumption that the
number of active secondary users are finite. In [15, 17, 22, 14, 20, 21, 23,
19], models with single channel are investigated.

In this paper, we first relax these assumptions by considering arbitrary
distributions for service time of primary users and secondary users. Under
these assumptions, we are able to obtain an explicit stability condition. To
the best of our knowledge, this is the first analytical result for cognitive
radio network models with multiple channels. Next, assuming exponential
service time distributions, we study the model in depth using the matrix
analytic method[32]. This allows us not only to recover the stability
condition but also to numerically evaluate effects of the input parameters on
the performance of secondary users. The advantage of the matrix analytic
method is to provide a systematic way to analyze more complicated models with
finite buffers for primary customers and/or Markovian arrival processes (MAP)
as well as Phase-type (PH) distributions [11]. The priority queues dominate
among models for service differentiation in communication networks and service
systems, and by this reason applications of our model are not restricted to
cognitive wireless networks [24].

The rest of this paper is organized as follows. In section 2, we describe our
model in detail with focus on the regenerative setting. In section 3, we
present the stability analysis of the basic model, while in section 3.1, as a
by product of our analysis, we derive the stability condition for multiserver
multiclass system without losses. Section 4 presents a detailed analysis for a
special case with exponential distributions for which we are able to obtain
performance measures. Finally, in Section 5, we present numerical and
simulation experiments to show insights into the performance of our system and
the sensitivity of service time distribution while Section 6 concludes our
paper.

##  2 Description of the system

We consider the following modification of the Erlang system with two classes
of customers with $c$ identical servers, Poisson inputs with rates
$\lambda_{i}$, general independent and identically distributed (i.i.d.)
service times $\\{S_{n}^{(i)},\,n\geq 1\\}$ for class-$i$ customers, $i=1,2$.
Class-1 customers have preemptive priority and are lost if meeting all busy
servers, while class-2 (non-priority) customers stay in the system regardless
of the state of the system and wait in the queue, if any, according to the
non-idling FCFS (first-come-first-served) service discipline. Class-1 and
class-2 customers correspond to primary and secondary customers while a server
is a channel, in cognitive radio networks. Denote service rates
$\mu_{i}=1/\hbox{\sf E}S^{(i)}$, where $S^{(i)}$ is the (generic) service time
of class-i customers. We also denote by $\\{t_{n},\,n\geq 1\\}$ the arrival
instants of the superposed (Poisson) input with rate
$\lambda=\lambda_{1}+\lambda_{2}$, and i.i.d. exponential interarrival times
$\\{\tau_{n}=t_{n+1}-t_{n}\\}$ with generic interarrival time $\tau$. By
preemptive-resume priority, a class-1 customer occupies a server busy by a
class-2 customer, provided there are no idle servers upon his arrival.

We deduce stability conditions of this system based on the regenerative
approach. More precisely, we show that the basic processes describing the
dynamics of the system are regenerative and then find conditions under which
these processes are positive recurrent [26, 25]. This approach, being quite
intuitive and transparent, is a powerful tool of performance and stability
analysis of a wide class of queueing processes including non-Markov ones as
well. The main idea of the regenerative stability analysis is based on a
characterization of the limiting remaining regeneration time. More precisely,
if we can show that this time does not go to infinity in probability, then the
mean regeneration period length is finite, and thus the process is positive
recurrent [28, 29].

First we describe the regenerative structure of the system. We denote by
$Q_{i}(t)$ the number of class-$i$ customers at time instant $t^{-}$ and let
$Q(t)=Q_{1}(t)+Q_{2}(t)$. Obviously

$\displaystyle Q_{1}(t)\leq c.$ (1)

Also let $W_{i}(t)$ be the workload (remaining work) at instant $t^{-}$ of
class-$i$ customers, $i=1,2$, and let $W(t)=W_{1}(t)+W_{2}(t)$. Denote by
$S_{i}(t)$ the remaining service time in server $i$ ($S_{i}(t)=0$ if the
server is idle), and let $R(t)$ be the set of servers occupied by class-1
customers at the instant $t$ (we put $R(t)=\emptyset$ if there are no such
customers). Then

$\displaystyle W_{1}(t)=\sum_{i\in R(t)}{S_{i}(t)}.$ (2)

Denote

$\displaystyle W_{n}=W(t_{n}),\quad Q_{n}=Q(t_{n}),$

so, at the arrival instant of customer $n$ the remaining work in all servers
equals $W_{n}$ and the total number of customers equals $Q_{n}$. Note that

$\\{W_{n}=0\\}=\\{Q_{n}=0\\},\,\,\,\,n\geq 1.$

Then the regeneration instants of the processes $\\{Q(t)\\}$ and $\\{W(t)\\}$
are defined as follows

$\displaystyle T_{n+1}=\min\\{t_{k}>T_{n}\,:\,W_{k}=Q_{k}=0\,\\},\,\,\,n\geq
0,\,\,\,T_{0}:=0,$ (3)

with generic regeneration period length $T$. It means that $T$ is distributed
as any distance between two regeneration points, i.e. as $T_{n+1}-T_{n}$. (To
define recursion (3) for $n\geq 1$, we put $T_{0}=0$ by definition.) To
explain, we note that a regeneration happens when a customer (of any class)
meets a completely idle system, and in this case the random instant $T_{n}$ is
the arrival instant of the $n$th customer who meets an idle system.

In what follows we assume that the first primary customer (of any class)
arrives in the idle system at instant $t_{1}=T_{0}=0$, in which case $t=0$ is
indeed the first (initial) regeneration point. It is called zero initial
state. We call regenerative process (in continuous-time case) positive
recurrent if the mean regeneration period length is finite, that is $\hbox{\sf
E}T<\infty.$ A key observation is that, because of the priority, the processes
$\\{Q_{1}(t)\\}$, $\\{W_{1}(t)\\}$ describing class-1 customers are
regenerative with regeneration instants

$T^{(1)}_{n+1}=\\{t_{n}^{(1)}>T_{n}^{(1)}:Q_{1}(t_{n}^{(1)})=0\\},\,n\geq
0\,\,(T^{(1)}_{0}=0),$

with generic regeneration period length $T^{(1)}$, where $\\{t_{n}^{(1)}\\}$
are the arrival instants of class-1 customers. In other words, $T_{n}^{(1)}$
is the $n$th arrival instant of class-1 customer which meets no other class-1
customers in the system. Because of (1), the process $\\{Q_{1}(t),\,t\geq
0\\}$ is tight, that is, for any $\varepsilon>0$, there exists a constant $C$
such that

$\inf_{t\geq 0}\hbox{\sf P}(Q_{1}(t)\leq C)\geq 1-\varepsilon.$

Then it follows from (2) and from the tightness of the process
$\\{S_{i}(t),\,t\geq 0\\}$ that the process $\\{W_{1}(t),\,t\geq 0\\}$ is
tight as well, see [27, 28]. It then follows from [29] that these processes
(describing class-1 customers solely) are also positive recurrent, that is

$\displaystyle\hbox{\sf E}\,T^{(1)}<\infty.$

Define the remaining regeneration time in the system at instant $t$ as

$\displaystyle T(t)=\min_{k}{(T_{k}-t:T_{k}-t\geq 0)},\,\,t\geq 0.$ (4)

In the regenerative stability analysis below we use the following basic result
from the renewal theory [30, 29]: if there exist a non-random sequence of time
instances $z_{i}\to\infty$, as $i\to\infty$, and constants $\delta>0$ and
$D<\infty$ such that

$\displaystyle\inf_{i}\hbox{\sf P}(T(z_{i})\leq D)\geq\delta,$ (5)

then $\hbox{\sf E}T<\infty$.

##  3 Stability analysis

We will prove conditions which imply positive recurrence of the system. On the
other hand, because the processes related to class-1 customers are positive
recurrent and the inputs are Poisson, in particular, there exists a stationary
probability $\mathsf{P}_{i}$ that exactly $i$ servers are busy by class-1
customers. In other words, there exist the limits

$\displaystyle\lim_{t\to\infty}\hbox{\sf P}(Q_{1}(t)=i)=\hbox{\sf
P}_{i},\,\,i=0,\ldots,c.$ (6)

Also denote $\rho_{2}=\lambda_{2}/\mu_{2}.$ It is assumed that the first
customer arrives in an empty system at instant $t_{1}=0$, and if the system is
empty at this instant, we call it zero initial state or initially empty
system. Note that in this case the instant $T_{0}=0$ is indeed a regeneration
point and then the first regeneration period is stochastically equivalent to
generic period, that is $T_{1}=_{st}T$. In this case positive recurrence means
that $\mathsf{E}T<\infty$. In this section we prove the following main result.

######  Theorem 3.1.

If condition

$\displaystyle\rho_{2}+\sum_{i=1}^{c}{i\mathsf{P}_{i}}<c,$ (7)

holds then $\mathsf{E}T<\infty$, that is initially empty system is positive
recurrent.

######  Proof 3.2.

In the interval of time $[0,t)$, denote: $\hat{V}_{1}(t)$ the work of class-1
customers accepted by the system (which means $\hat{V}_{1}(t)$ does not
include the lost work); $V_{2}(t)$ the received work of class-2 customers;
$B(t)$ the aggregated busy time of the servers which equals departed work in
$[0,t)$. Also denote $I(t)=\sum_{i=1}^{c}{I_{i}(t)}$, where $I_{i}(t)$ is the
idle time of server $i$ in $[0,\,t]$, so $B(t)=ct-I(t)$. It is now clear that
the following balance equation holds.

$\displaystyle\hat{V}_{1}(t)+V_{2}(t)=W_{1}(t)+W_{2}(t)+B(t)=W_{1}(t)+W_{2}(t)+ct-I(t).$
(8)

First of all we note that, by positive recurrence of ”class-1 processes”, we
obtain (see [25]) that

$\displaystyle\lim_{t\to\infty}\frac{1}{t}\hbox{\sf
E}Q_{1}(t)=0,\quad\lim_{t\to\infty}\frac{1}{t}\hbox{\sf E}W_{1}(t)=0.$ (9)

Note that the convergence with probability 1 (w.p.1), that is,

$Q_{1}(t)=o(t),\,\,\,W_{1}(t)=o(t),\,\,t\to\infty,$

also holds in (9). Now we apply a proof by contradiction and assume that,
under condition (7),

$\displaystyle Q_{2}(t)\Rightarrow\infty,\,\,\,t\to\infty,$ (10)

which means the queue size of class-2 customers increases infinitely in
probability. It means that, for each fixed $k$,

$\displaystyle\lim_{t\to\infty}\mathsf{P}(Q_{2}(t)>k)=1.$ (11)

Denote by $\hbox{\sf 1}(\cdot)$ the indicator function, then

$I_{i}(t)=\int_{0}^{t}\hbox{\sf 1}(S_{i}(u)=0)du,\,\,\,t\geq 0,$

is the idle time of server $i$ in the interval $[0,\,t]$. It then follows that
the average aggregated idle time of all servers in the interval of time
$[0,\,t]$ is

$\displaystyle\hbox{\sf E}I(t)=\hbox{\sf
E}\Big{[}\sum_{i=1}^{c}\int_{0}^{t}\hbox{\sf
1}(S_{i}(u)=0)du\Big{]}=\sum_{i=1}^{c}\int_{0}^{t}\hbox{\sf P}(S_{i}(u)=0)du.$

Because the service discipline is non-idling, then

$\displaystyle\mathsf{P}(Q_{2}(t)>c)\leq\mathsf{P}(S_{i}(t)>0),\,\,\,i=1,\ldots,c,$
(12)

and, by (11), (12), we obtain as $t\to\infty$,

$\displaystyle\hbox{\sf P}(S_{i}(t)=0)=1-\hbox{\sf P}(S_{i}(t)>0)\leq
1-\hbox{\sf P}(Q_{2}(t)>c)\rightarrow 0.$

It now easily follows that

$\displaystyle\lim_{t\to\infty}\frac{1}{t}\hbox{\sf
E}I(t)=\lim_{t\to\infty}\sum_{i=1}^{c}\frac{1}{t}\int_{0}^{t}\hbox{\sf
P}(S_{i}(u)=0)du=0.$ (13)

Denote by $A_{2}(t)$ the number of class-2 customers arrived in the interval
$[0,t)$. The process $\\{A_{2}(t),\,t\geq 0\\}$ is a positive recurrent
cumulative process [25] or, equivalently, the process with regenerative
increments [31] in which each arrival instant of class-2 customer is a
regeneration instant of the incoming load process

$V_{2}(t)=\sum_{k=1}^{A_{2}(t)}S_{k}^{(2)},\,\,t\geq 0.$

We note that the i.i.d. exponential interarrival times $\\{\tau_{n}^{(2)}\\}$
between class-2 customers are the regeneration periods of the load process
$\\{V_{2}(t),\,t\geq 0\\}$. Denote the generic period $\tau^{(2)}$ which has
parameter $\lambda_{2}$. Also we note that the process $\\{V_{2}(t)\\}$ has
the increment $S^{(2)}$ over interval $\tau^{(2)}$. Then it follows, for
instance, from Theorem 55 in [31], that

$\displaystyle\lim_{t\to\infty}\frac{\mathsf{E}V_{2}(t)}{t}=\frac{\hbox{\sf
E}S^{(2)}}{\hbox{\sf E}\tau^{(2)}}=\rho_{2}.$ (14)

Now we define the total busy time $B_{1}(t)$ when all servers are occupied by
class-1 customers, in the interval $[0,\,t]$. A key observation is that
$B_{1}(t)$ can be represented as follows

$\displaystyle B_{1}(t)$ $\displaystyle=$ $\displaystyle\int_{0}^{t}\hbox{\sf
1}(Q_{1}(u)=1)du+2\int_{0}^{t}\hbox{\sf 1}(Q_{1}(u)=2)du+\cdots$
$\displaystyle+$ $\displaystyle c\int_{0}^{t}\hbox{\sf
1}(Q_{1}(u)=c)du=\sum_{i=1}^{c}i\int_{0}^{t}\hbox{\sf 1}(Q_{1}(u)=i)du.$

This gives the following equality

$\displaystyle\hat{V}_{1}(t)=B_{1}(t)+W_{1}(t)=\sum_{i=1}^{c}i\int_{0}^{t}\hbox{\sf
1}(Q_{1}(u)=i)du+W_{1}(t),\,\,\,t\geq 0.$

Recall that all processes related to class-1 customers are positive recurrent
regenerative. Moreover, since the input is Poisson, the weak limit
$Q_{1}(t)\Rightarrow Q_{1}$ exists and is the stationary number of servers
occupied by class-1 customers with distribution (6). It is worth mentioning
that, by the property PASTA [26], the stationary probability $\hbox{\sf
P}_{i}$ is also the limiting fraction of the time when there are exactly $i$
class-1 customers in the system, which means

$\displaystyle\mathsf{P}_{i}=\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}{\hbox{\sf
P}(Q_{1}(u)=i)du},\,\,i=1,\ldots,c.$ (15)

Now it follows from (9), (19), (15) that

$\displaystyle\lim_{t\to\infty}\frac{1}{t}\hbox{\sf
E}\hat{V}_{1}(t)=\sum_{i=1}^{c}i\lim_{t\rightarrow\infty}\frac{1}{t}\int_{0}^{t}{\mathsf{P}(Q_{1}(u)=i)du}=\sum_{i=1}^{c}i\mathsf{P}_{i}.$
(16)

Now collecting together results (9), (13), (14) and (16), we obtain from the
balance equation (8) that

$\displaystyle\rho_{2}+\sum_{i=1}^{c}i\hbox{\sf
P}_{i}=\lim_{t\to\infty}\frac{1}{t}\hbox{\sf E}W_{2}(t)+c,$

implying

$\displaystyle\lim_{t\rightarrow\infty}\frac{1}{t}\hbox{\sf
E}W_{2}(t)=\sum_{i=1}^{c}i\hbox{\sf P}_{i}+\rho_{2}-c\geq 0,$

or

$\displaystyle\rho_{2}+\sum_{i=1}^{c}i\hbox{\sf P}_{i}\geq c.$

This contradiction with assumption (7) shows that assumption (10) is false,
and thus $Q_{2}(t)\not\Rightarrow\infty$. Hence there exist constants
$N<\infty,\,\delta_{0}>0$ and a (non-random) sequence $u_{i}\to\infty$ such
that (cf. (5))

$\displaystyle\inf_{i}\mathsf{P}(Q_{2}(u_{i})\leq N)\geq\delta_{0}.$ (17)

Note that

$W_{2}(t)\leq\sum_{k=1}^{Q_{2}(t)}S_{k}^{(2)}+\sum_{k=1}^{c}S_{k}(t),\,\,\,t\geq
0,$

and that the remaining service times $\\{S_{k}(t),\,t\geq 0\\}$ are the tight
processes $k=1,\ldots,c$, see [27]. A routine but tedious calculation confirms
an intuitive result, that the bound (17) implies the corresponding lower bound
for the workload process $\\{W_{2}(t),\,t\geq 0\\}$, namely

$\displaystyle\inf_{i}\mathsf{P}(W_{2}(u_{i})\leq D_{2})\geq\delta_{2},$

for some constants $D_{2},\,\delta_{2}>0$. Moreover, because the positive
recurrent process $\\{W_{1}(t),\,t\geq 0\\}$ is also tight, in particular,

$\displaystyle\inf_{i}\mathsf{P}(W_{2}(u_{i})\leq D_{2},\,W_{1}(u_{i})\leq
D_{1})\geq\delta_{1},$

for some constants $D_{1},\,\delta_{1}>0$. Denote $\tau(t)$ the remaining
(exponential) interarrival time at instant $t$ (in the superposed Poisson
input with rate $\lambda$), so $\hbox{\sf P}(\tau(t)\leq x)\geq e^{-\lambda
x}$ for any $x\geq 0$. Recall definition (4) and note that, for arbitrary
fixed $u_{i}$ (satisfying (17)) and a constant $L>D_{1}+D_{2}$, the following
lower bound for the remaining regeneration time holds.

$\displaystyle\hbox{\sf P}(T(u_{i})\leq L)$ $\displaystyle\geq$
$\displaystyle\hbox{\sf P}\Big{(}W_{2}(u_{i})\leq D_{2},\;W_{1}(u_{i})\leq
D_{1},\,L\geq\tau(u_{i})>D_{1}+D_{2}\Big{)}$ $\displaystyle\geq$
$\displaystyle\delta_{1}(e^{-\lambda(D_{1}+D_{2})}-e^{-\lambda
L})=:\varepsilon>0.$

To explain this inequality, we note that on the event

$\Big{\\{}W_{2}(u_{i})\leq D_{2},\;W_{1}(u_{i})\leq
D_{1},\,L\geq\tau(u_{i})>D_{1}+D_{2}\Big{\\}},$

the first customer arriving after instant $u_{i}$ (at instant
$u_{i}+\tau(u_{i})\leq u_{i}+L$) meets a completely idle system and thus a
regeneration occurs. It explains why then $T(u_{i})\leq L$. Because this bound
is uniform in $u_{i}$ and $i$, we obtain that the inequality (5) holds and
thus $\hbox{\sf E}T<\infty$. Thus we conclude that (7) is the sufficient
stability condition.

Now we show that (7) is the necessary stability condition as well. Namely, we
assume that the initially empty system is positive recurrent, which means
$\mathsf{E}T<\infty$. Note that

$I(t)\geq\int_{0}^{t}\hbox{\sf 1}(Q(u)=0)du=:I_{0}(t),$

where $I_{0}(t)$ is the aggregated time when all servers are simultaneously
free in the interval $[0,\,t]$. Then it follows from the theory of
regenerative processes that

$\displaystyle\lim_{t\to\infty}\frac{I(t)}{t}\geq\lim_{t\to\infty}\frac{I_{0}(t)}{t}=\frac{\hbox{\sf
E}I_{0}}{\hbox{\sf E}T},$ (18)

where $I_{0}$ is the idle time, during a regeneration period, when all servers
are simultaneously free [26, 31]. Because $\tau$ is exponential and $\hbox{\sf
E}S^{(i)}<\infty$, for arbitrary $\delta>0$ and some $\varepsilon_{1}>0$,

$\mathsf{P}(\tau>\delta+S^{(i)})=\varepsilon_{1},\,\,\,i=1,2.$

Hence,

$\mathsf{E}I_{0}\geq\mathsf{E}(I_{0};\tau-S^{(i)}>\delta)\geq\delta\delta_{1}>0,$

and thus the limit in (18) is positive. It then immediately follows from (14),
(16), (18), and from the balance equation (8), by dividing both sides by $t$
and letting $t\to\infty$, that the inequality (7) holds. This implies that (7)
is indeed the necessary stability condition. Combining this result with the
statement of Theorem 3.1 we obtain the following main statement.

######  Theorem 3.3.

The initially empty system under consideration is positive recurrent if and
only if condition (7) holds.

######  Remark 3.4.

Following [29] one can extend the analysis to prove that, under condition (7),
the system is positive recurrent under arbitrary fixed values $W(0)$ and
$Q(0)$. In this case we shall also show that the 1st regeneration period,
which in general has another distribution than $T$, is finite w.p.1.

It is worth mentioning that, because to analyse class-1 customers only, we can
treat the system as a loss $M/G/c/0$ system, then the stationary distribution
$\\{\mathsf{P}_{i}\\}$ can be found from the celebrated Erlang forula, for
instance, see [26].

Denote by $\zeta$ the stationary number of servers available for class-2
customers, that is (stochastically),

$\zeta=_{st}c-\sum_{i=1}^{c}i\hbox{\sf 1}(Q_{1}=i).$

Now, denoting $\Delta=\hbox{\sf E}\zeta$, we obtain that the mean stationary
number of servers available for class-2 customers is

$\Delta=c-\sum_{i=1}^{c}i\mathsf{P}_{i}.$

Then condition (7), rewritten as

$\displaystyle\rho_{2}<\Delta,$ (19)

has the following probabilistic interpretation: the traffic intensity of
class-2 customers must be less than the mean number of the available servers
$\Delta$. Condition (19) is a negative drift condition and, being intuitive,
however requires a strict proof because, unlike classic systems with a fixed
number of servers, in this case the number of available servers $\zeta$ is
random. This is the main reason why we present above the detailed regenerative
proof of this result.

###  3.1 $M/G/c$ system without losses

In this section, keeping previous notation, we obtain as a by-product of our
analysis, a well-known stability condition of a buffered multiclass system,
that is the system with the infinite capacity buffer for both classes of
customers. Thus the only difference between the original and the new systems
is that there are no losses in the new system.

######  Theorem 3.5.

The initially empty system without losses is positive recurrent if and only if
the condition

$\rho_{1}+\rho_{2}<c$

holds.

The proof of this statement is quite similar to that has been given in the
previous section. The only difference is that now we replace $\hat{V}_{1}(t)$
by the full work $V_{1}(t)$ generated by all class-1 customers which arrive in
the interval of time $[0,\,t)$. It remains to note that

$\lim_{t\to\infty}\frac{\hbox{\sf E}V_{1}(t)}{t}=\rho_{1}.$

######  Remark 3.6.

Using previous arguments, it is straightforward to show that condition

$\sum_{i=1}^{N}{\rho_{i}}<c$

is stability criterion for $N$-class system with no losses and arbitrary work-
conserving discipline including various priority policies.

##  4 Exponential service time distributions

In this section, we examine more closely a special case of the system
described in Section 2, in which service times of both customer classes follow
exponential distributions, with parameter $\mu_{i}$ for class-$i$ customers.

In this pure Markovian case, for stability analysis it is possible by applying
the matrix analytic method [32]. This alternative proof of stability is
instructive. Moreover, in this setting we can calculate the stationary
distribution of the basic Markov process and as a result obtain the
corresponding stationary performance indexes. In this setting, the process
$\\{(Q_{1}(t),Q_{2}(t)),\,t\geq 0\\}$ is a continuous-time Markov Chain with
the state space $\mathbb{S}$ given by

$\mathbb{S}=\\{(i,j)\in\\{0,1,...,c\\}\times\mathbb{N}\\}.$

Grouping states into levels according to their values of $Q_{2}(t)$, the
system can be formulated as a Quasi-Birth-and-Death process (QBD) whose the
infinitesimal generator $Q$ is expressed as follows

$Q=\begin{pmatrix}B_{0}&C&O&O&O&...&...&...&...\\\
A_{1}&B_{1}&C&O&O&...&\vdots&\vdots&\vdots\\\
O&A_{2}&B_{2}&C&O&...&\vdots&\vdots&\vdots\\\
\vdots&\vdots&\ddots&\ddots&\ddots&\vdots&\vdots&\vdots&\vdots\\\
O&O&...&A_{c}&B_{c}&C&O&...&\vdots\\\ O&O&...&O&A_{c}&B_{c}&C&O&...\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\vdots&\\\
\end{pmatrix},$

where $O$ is a $(c+1)\times(c+1)$ zero matrix, and $A_{n},B_{n},C$ are
$(c+1)\times(c+1)$ block matrices given by

$C=\begin{pmatrix}\lambda_{2}&0&0&0\\\ 0&\lambda_{2}&0&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&\lambda_{2}\end{pmatrix},$

$A_{n}=\begin{pmatrix}n\mu_{2}&0&0&...&0&0&...&0\\\
0&n\mu_{2}&0&...&0&0&...&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&...&n\mu_{2}&0&...&0&0\\\ 0&0&...&0&(n-1)\mu_{2}&...&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&...&0&0&...&\mu_{2}&0\\\ 0&0&...&0&0&...&0&0\end{pmatrix},$

for $n\leq c$, and $A_{n}=A_{c}$, for $n>c$;

$B_{n}=\begin{pmatrix}b_{n,0}&\lambda_{1}&0&...&0&0&...&0&0\\\
\mu_{1}&b_{n,1}&\lambda_{1}&...&0&0&...&0&0\\\
0&2\mu_{1}&b_{n,2}&\ddots&0&0&...&0&0\\\
\vdots&\vdots&\ddots&\ddots&\ddots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&...&b_{n,c-n}&\lambda_{1}&...&0&0\\\
0&0&0&...&(n-1)\mu_{1}&b_{n,c-(n-1)}&\ddots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\vdots\\\
0&0&0&...&0&0&...&b_{n,c-1}&\lambda_{1}\\\
0&0&0&...&0&0&...&c\mu_{1}&-(\mathrel{{\lambda_{2}}{+}{c\mu_{1}}})\\\
\end{pmatrix},$

for $n\leq c$, and $B_{n}=B_{c}$ for $n>c$. In case $n\leq c$, the diagonal
elements of $B_{n}$ are given by

$b_{n,i}=-(\lambda_{1}+\lambda_{2}+i\mu_{1}+min\\{c-i,n\\}\mu_{2}).$

Denote by $\boldsymbol{\eta}=(\eta_{0},\eta_{1},...,\eta_{c})$ the row vector
representing the stationary distribution of the infinitesimal generator
$Q^{*}=A_{c}+B_{c}+C$. It is easy to see that

$\boldsymbol{\eta}=\left(\eta_{0},\rho_{1}\eta_{0},\frac{\rho_{1}^{2}}{2!}\eta_{0},...,\frac{\rho_{1}^{i}}{i!}\eta_{0},...,\frac{\rho_{1}^{c}}{c!}\eta_{0}\right),$

where

$\eta_{0}=\frac{1}{\sum_{i=0}^{c}\frac{\rho_{1}^{i}}{i!}}\,\,\,\mbox{and}\,\,\,\rho_{1}=\frac{\lambda_{1}}{\mu_{1}}.$

The Quasi-Birth-Death process is ergodic if and only if the following
condition holds [32].

$\boldsymbol{\eta}C\boldsymbol{e}<\boldsymbol{\eta}A_{c}\boldsymbol{e},$

where $\boldsymbol{e}$ denotes the $(c+1)$-dimension column vector of ones.
This stability condition can be further transformed as

$\displaystyle\lambda_{2}<\frac{\sum_{i=0}^{c-1}(c-i)\frac{\rho_{1}^{i}}{i!}}{\sum_{i=0}^{c}\frac{\rho_{1}^{i}}{i!}}\mu_{2}.$
(20)

The above condition can also be rewritten as

$\rho_{2}+\sum_{i=1}^{c}i\eta_{i}<c,$

which is identical to the stability condition in general case stated in (7).

In what follows, we consider the system under the stability condition. Now,
let $\pi(i,j)=\hbox{\sf P}(Q_{1}(t)=i,Q_{2}(t)=j)$ denote the stationary
probability of the Markov chain, and let

$\boldsymbol{\pi}_{j}=(\pi(0,j),\pi(1,j),...,\pi(c,j))\,\,\,\mbox{for}\,\,\,j\in\mathbb{N}.$

According to Matrix-analytic-method [32, 33], we have

$\boldsymbol{\pi}_{j}=\boldsymbol{\pi}_{c}R^{j-c},\quad j\geq
c,\qquad\boldsymbol{\pi}_{j}=\boldsymbol{\pi}_{j-1}R^{(j)},\quad
j=c,c-1,\dots,1,$

where $R$ is the minimal non-negative solution of

$C+RB_{c}+R^{2}A_{c}=O,$

and

$R^{(j)}=-C(B_{j}+R^{(j+1)}A_{j+1})^{-1},\quad j=c-1,c-2,\dots,1,$

given that $R^{(c)}=R$ and $R$ is numerically computed using algorithms in
[32].

Finally, $\boldsymbol{\pi}_{0}$ is the unique solution of the following
equations

$\displaystyle\boldsymbol{\pi}_{0}(B_{0}+R^{(1)}A_{1})$
$\displaystyle=\boldsymbol{0},$
$\displaystyle\boldsymbol{\pi}_{0}\left(I+\sum_{i=1}^{c-1}\prod_{j=1}^{i}R^{(j)}+\left(\prod_{j=1}^{c}R^{(j)}\right)(I-R)^{-1}\right)\boldsymbol{e}$
$\displaystyle=1,$

where $I$ denotes the $(c+1)\times(c+1)$ identity matrix.

Moreover, due to Little’s law, we obtain the average waiting time,
$\mathsf{E}W_{q}$, of class-2 customers as follows

$\displaystyle\mathsf{E}W_{q}$
$\displaystyle=\frac{\sum_{j=0}^{\infty}\sum_{i=0}^{c}\max\\{0,i+j-c\\}\pi(i,j)}{\lambda_{2}}$
$\displaystyle=\frac{\sum_{j=1}^{c-1}\sum_{i=c-j+1}^{c}(i+j-c)\pi(i,j)+\boldsymbol{\pi}_{c}(I-R)^{-2}R\boldsymbol{e}+\boldsymbol{\pi}_{c}(I-R)^{-1}\boldsymbol{f}}{\lambda_{2}},$

where $\boldsymbol{f}=(0,1,2,...,c-1,c)^{T}$.

There are other performance measures which describe QoS of such a system. For
instance, if a class-2 customer is interrupted by class-1 customer, we call it
a termination event. Denote the set

$\mathbb{S}^{*}=\\{(i,j):i+j\geq c,\,i\leq c-1\\},$

which contains the states when all servers are occupied and there is at least
one class-2 customer occupying a server. Then the average number of
termination events per class-2 customer is given by

$\mathsf{E}N_{T}=\frac{\lambda_{1}}{\lambda_{2}}\sum_{(i,j)\in\mathbb{S}^{*}}\pi(i,j),$

where we apply the property PASTA [26]. We also present a new alternative way
how to calculate $\mathsf{E}N_{T}$ based on the regenerative arguments.
Indeed, denote by $A_{1}(t)$ the number of class-1 customers arriving in the
interval $[0,\,t)$ and recall similar definition $A_{2}(t)$ for class-2
arrivals. Recall notation

$Q(t)=Q_{1}(t)+Q_{2}(t)\Rightarrow Q=Q_{1}+Q_{2}.$

Then using the basic asymptotic results for positive recurrent regenerative
processes [26, 29], we obtain $\mathsf{E}N_{T}$ as the following w.p.1 limit
of fraction of class-1 arrivals interrupting class-2 customers:

$\displaystyle\mathsf{E}N_{T}$ $\displaystyle=$
$\displaystyle\lim_{t\to\infty}\frac{1}{A_{2}(t)}\sum_{n=1}^{A_{1}(t)}\hbox{\sf
1}(Q(t_{n}^{(1)})\geq c,\,Q_{1}(t_{n}^{(1)})\leq c-1)$ (21) $\displaystyle=$
$\displaystyle\lim_{t\to\infty}\frac{A_{1}(t)}{t}\frac{1}{A_{1}(t)}\sum_{n=1}^{A_{1}(t)}\hbox{\sf
1}(Q(t_{n}^{(1)})\geq c,\,Q_{1}(t_{n}^{(1)})\leq
c-1)\cdot\lim_{t\to\infty}\frac{t}{A_{2}(t)}$ $\displaystyle=$
$\displaystyle\lambda_{1}\mathsf{P}(Q\geq c,\,Q_{1}\leq
c-1)\frac{1}{\lambda_{2}}=\frac{\lambda_{1}}{\lambda_{2}}\sum_{(i,j)\in\mathbb{S}^{*}}\mathsf{P}(Q_{1}+Q_{2}=i+j,\,Q_{2}=j)$
$\displaystyle=$
$\displaystyle\frac{\lambda_{1}}{\lambda_{2}}\sum_{(i,j)\in\mathbb{S}^{*}}\pi(i,j),$

where we also use property PASTA to apply the equality

$\pi(i,j)=\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\sum_{(i,j)\in\mathbb{S}^{*}}\hbox{\sf
1}(Q(t_{k}^{(1)})=i+j,\,Q_{2}(t_{k}^{(1)})=j).$

The equality (21) written as

$\lambda_{2}\mathsf{E}N_{T}=\lambda_{1}\sum_{(i,j)\in\mathbb{S}^{*}}\pi(i,j)$

is intuitive and establishes a balance between the rate of the interrupted
class-2 customers and the rate of interrupting class-1 customers.

######  Remark 4.1.

It is worth mentioning that relation (21) holds also for general service time
distribution of any class of customers, however in this case the stationary
distribution $\\{\pi(i,j)\\}$ of (non-Markovian process
$\\{(Q_{1}(t),\,Q_{2}(t)),\,t\geq 0\\}$) is not analytically available.

##  5 Simulations and Numerical Insights

In this section, we present some numerical examples of the results obtained by
the matrix analytic method presented in Section 4. In our experiment, for
fixed $\mu_{1}=4,\mu_{2}=20$ and values $c=2,5$, we show the changes in the
values of some performance measures against $\lambda_{1}$ and $\lambda_{2}$.
Under the same settings, we also carry out simulations and obtain the same
results as those obtained by the matrix analytic method. Furthermore, to show
the sensitivity of the service time distribution of primary users, we also
compare the results by matrix analytic methods with those by simulations where
service time of class-1 customers follows Erlang distributions with the shape
parameter $r=5,10$.

The scale parameter is chosen such that the mean value remains the same as in
the case of exponential distributions. The duration for all experimental
simulations is set at $10^{6}$ time units, which is adequate for the
simulation results to converge to their corresponding numerical results. The
simulation results in all the figures are represented by the points marked
with notation $\mathsf{sim}$, without which the results are understood to be
obtained from numerical calculations. Simulation results with service time of
class-1 customers following Erlang distributions are marked with the
abbreviation $\mathsf{Erl.}$ in the legend.

Figure 1: Average waiting time of class-2 customers against arrival rate of
class-1 customers $(\lambda_{2}=8)$. Figure 2: Average waiting time of class-2
customers against arrival rate of class-2 customers ($\lambda_{1}=1$).

Figure 1 and Figure 2 compare average waiting time of class-2 customers as
$\lambda_{1},\lambda_{2}$ and $c$ vary. It can be seen that the average amount
of time that class-2 customers spend in the system ($\mathsf{E}W_{q}$) is
larger when customers of either class arrive more frequently, or when there
are fewer servers. We observe that $\mathsf{E}W_{q}$ with exponential service
time distribution for class-1 is largest while $\mathsf{E}W_{q}$ with $r=5$ is
larger than that with $r=10$. This indicates that $\mathsf{E}W_{q}$ increases
with the increase in the variance of service time of class-1 customers.

Figure 3 illustrates how the average number of termination events per class-2
customer ($\mathsf{E}N_{T}$) changes according to the arrival rates of class-1
customers and the number of servers. Obviously, when class-1 customers arrive
more frequently, more class-2 customers’ sessions are terminated. Also, as the
number of servers is larger, there are more spaces for all customers, and thus
class-2 customers are less likely to be kicked out of the servers. The same
patterns can be observed in Figure 4. Furthermore, simulations results for
Erlang service time distributions of class-1 customers are almost the same as
the corresponding ones with exponential distributions. This shows that
$\mathsf{E}N_{T}$ is (almost) insensitive to the service time distribution of
class-1 customers.

Figure 3: Average number of termination events per class-2 customers against
arrival rate of class-1 customers $(\lambda_{2}=8)$. Figure 4: Average number
of termination events per class-2 customers against arrival rate of class-2
customers ($\lambda_{1}=1$).

Figure 5 shows the distributions of the number of times that each class-2
customer is terminated. In this experiment, we set
$\lambda_{1}=20,\lambda_{2}=30,c=10$, and let the service time of class-1
customers follow an exponential distribution (denoted by $\mathsf{Exp.}$ in
the figure), and Erlang distributions. It can be seen that higher numbers of
termination times occur with smaller probabilities. Also, there is no
significant difference in the results when we modify the distribution of
service time for class-1 customers under the three settings.

Figure 5: Distribution of number of termination events
($\lambda_{1}=20,\lambda_{2}=30,c=10$). Figure 6: Throughput against arrival
rate of class-1 customers ($\lambda_{2}=8$). Figure 7: Throughput against
arrival rate of class-2 customers ($\lambda_{1}=1$).

Figure 6 reflects changes in the throughput of class-2 customers against
$\lambda_{1}$ when $\lambda_{2}$ is fixed. It can be seen that the throughput
values remain unchanged at $\lambda_{2}$ when $\lambda_{1}$ goes up to certain
thresholds, then drop as $\lambda_{1}$ continues to increase. Meanwhile,
Figure 7 indicates that the throughput is equal to $\lambda_{2}$ up to a
certain threshold of $\lambda_{2}$, then remains unchanged at a value
$\lambda_{max}$ when class-2 customers arrive at very high rates. At that,
$\lambda_{max}$ is defined as

$\lambda_{max}=\frac{\sum_{i=0}^{c-1}(c-i)\frac{\rho_{1}^{i}}{i!}}{\sum_{i=0}^{c}\frac{\rho_{1}^{i}}{i!}}\mu_{2},$

which is the right hand side of the stability condition (20).

Finally, it is noticeable that when we let service times of class-1 customers
follow Erlang distributions, the results do not change much as compared to the
exponential case. This agrees with our stability condition which depends on
only the mean service time of class-1 customers.

##  6 Conclusion

In this paper, we have considered a modified Erlang system for cognitive radio
networks and related applications. We have established a stability condition
which is insensitive to the service time distributions of primary and
secondary users. This result has implied that the throughput of secondary
users is insensitive to the service time distributions of primary and
secondary users, provided that the means are fixed. For the case of
exponential service time distributions of primary and secondary customers, we
have derived some stationary performance measures. Our extensive simulations
have shown that the mean waiting time of secondary users increases with the
increase in the variance of service time of primary users while the mean
number of terminations of secondary users is almost insensitive to the service
time distributions of primary users, provided that the means are fixed. Our
findings could be used in resource allocation in cognitive radio networks and
related applications.

The research of EM and SR is supported by the Russian Foundation for Basic
Research, projects 18-07-00147, 18-07-00156 and 19-07-00303. The research of
TP and HQN was supported in part by JSPS KAKENHI Grant Number 18K18006.

## References

  * [1] Akyildiz, I. F.; Lee, W.-Y.; Vuran, M. C.; Mohanty, S. NeXt generation dynamic spectrum access cognitive radio wireless networks: a survey. Comput. Netw. 2006, 50(13), 2127–2159.
  * [2] Akyildiz, I. F.; Lee, W. Y.; Vuran, M. C.; Mohanty; S. A survey on spectrum management in cognitive radio networks. IEEE Commun. Mag. 2008, 46(4), 40-48.
  * [3] Letaief, K. B.; Zhang, W. Cooperative communications for cognitive radio networks. Proc. IEEE. 2009, 97(5), 878-893.
  * [4] Mitola, J.; Maguire, G. Q. Cognitive radio: making software radios more personal. IEEE Pers. Commun. 1999, 6(4), 13-18.
  * [5] Wang, B.; Liu, K. R. Advances in cognitive radio networks: A survey. IEEE J. Sel. Top. Signal Process. 2010, 5(1), 5-23.
  * [6] Ostovar, A.; Keshavarz, H.; Quan, Z. Cognitive radio networks for green wireless communications: an overview. Telecommun. Syst. 2020, 1-10.
  * [7] Mitrany, I. L.; Avi-Itzhak, B. A many-server queue with service interruptions. Oper. Res. 1968, 16(3), 628-638.
  * [8] Akutsu, K.; Phung-Duc, T. Analysis of retrial queues for cognitive wireless networks with sensing time of secondary users, Lect. Notes Comput. Sci. 2019, LNCS 11688, 77-91.
  * [9] Salameh, O.; Bruneel, H.; Wittevrongel, S. Performance evaluation of cognitive radio networks with imperfect spectrum sensing and bursty primary user traffic. Math. Probl. Eng. 2020\.
  * [10] Salameh, O.; De Turck, K.; Bruneel, H.; Blondia; C.; Wittevrongel, S. Analysis of secondary user performance in cognitive radio networks with reactive spectrum handoff. Telecommun. Syst. 2017, 65(3), 539-550.
  * [11] Dudin, A. N.; Lee, M. H.; Dudina, O.; Lee, S. K. Analysis of priority retrial queue with many types of customers and servers reservation as a model of cognitive radio system. IEEE Trans. Commun. 2016, 65, 186-199.
  * [12] Konishi, Y.; Masuyama, H.; Kasahara, S.; Takahashi, Y. Performance analysis of dynamic spectrum handoff scheme with variable bandwidth demand of secondary users for cognitive radio networks. Wirel. Netw. 19\. 2013, 607–617.
  * [13] Salameh, O.; De Turck, K.; Bruneel, H.; Blondia, C.; Wittevrongel, S. Analysis of secondary user performance in cognitive radio networks with reactive spectrum handoff. Telecommun. Syst. 2017, 65, 539–550.
  * [14] Paul, S.; Phung-Duc, T. Retrial queueing model with two-way communication, unreliable server and resume of interrupted call for cognitive radio networks. In Information technologies and mathematical modelling. Queueing theory and applications, Springer, Cham. 2018, pp. 213-224.
  * [15] Shajin, D.; Dudin, A. N.; Dudina, O.; Krishnamoorthy, A. A two-priority single server retrial queue with additional items. J. Ind. Manag. Optim. 2019\.
  * [16] Wong, E. W. A.; Foh, C. H. Analysis of cognitive radio spectrum access with finite user population. IEEE Wireless Commun. Lett. 2009, 13(5), 294–296.
  * [17] Zhang, Y.; Wang, J. ; Li, W. W. Optimal pricing strategies in cognitive radio networks with heterogeneous secondary users and retrials. IEEE Access. 2019, 7, 30937-30950.
  * [18] Liu, J.; Jin, S.; Yue, W. Performance evaluation and system optimization of Green cognitive radio networks with a multiple-sleep mode. Ann. Oper. Res. 2019, 277, 371-391.
  * [19] Azarfar, A.; Frigon, J. F.; Sanso, B. Priority queueing models for cognitive radio networks with traffic differentiation. EURASIP J Wirel. Commun. Netw. 2014, 206.
  * [20] Dimitriou, I.; Phung-Duc, T. Analysis of cognitive radio networks with cooperative communication. Proceedings of the 13th EAI International Conference on Performance Evaluation Methodologies and Tools, 2020; pp. 192-195.
  * [21] Dragieva, V. I.; Phung-Duc, T. Queueing analysis of cognitive radio networks with finite number of secondary users. Proceedings of the 25th International Conference on Analytical and Stochastic Modelling Techniques and Applications, 2019.
  * [22] Nazarov, A.; Phung-Duc, T.; Paul, S. Unreliable single-server queue with two-Way communication and retrials of blocked and interrupted calls for cognitive radio networks. Distributed Computer and Communication Networks, pp. 276-287 (2018), Springer, Cham.
  * [23] Kim, K. T-preemptive priority queue and its application to the analysis of an opportunistic spectrum access in cognitive radio networks. Comput. Oper. Res. 2012, 39(7), 1394-1401.
  * [24] Vishnevskiy, V. M.; Samouylov, K. E.; Yarkina, N. V. Mathematical model of LTE cells with machine-type and broadband communications. Autom. Remote Control. 2020, 81, 622-636.
  * [25] Smith, W.L. Regenerative stochastic processes. Proc. R. Soc. Lond. A (232), 6-31 (1955).
  * [26] Asmussen, S. Applied probability and Queues. 2nd edn. Springer, Springer-Verlag New York (2003).
  * [27] Morozov, E. The tightness in the ergodic analysis of regenerative queueing processes. Queueing Syst. 1997, 27, 179-203.
  * [28] Morozov, E. Weak regeneration in modeling of queueing processes. Queueing Syst. 2004, 46, 295-315.
  * [29] Morozov E., Delgado R. Stability analysis of regenerative queues, Autom. Remote Control. 2009, 70, 1977-1991.
  * [30] Feller, W. An introduction to probability theory vol.2, Wiley, New York (1971).
  * [31] Serfozo, R. Basics of applied stochastic processes, Springer-Verlag (2009).
  * [32] Neuts, M.F. Matrix geometric solutions in stochastic models—an algorithmic approach. Johns Hopkins University Press, Baltimore, MD (1981).
  * [33] Phung-Duc, T.; Masuyama, H.; Kasahara, S.; Takahashi, Y. A simple algorithm for the rate matrices of level-dependent QBD processes. Proceedings of the 5th International Conference on Queueing Theory and Network Applications, 2010; pp. 46-52.