Queueing Analysis of Cellular Mobile Radio System Based on Prioritized
Channel Assignment
M. Jain†, Rakhee+ and Kriti Priya*
†
+
School of Mathematical Sciences, Institute of Basic Science, Khandari, Agra-282002 (INDIA).
Apaji Institute of Mathematics and Applied Computer Technology, Banasthali Vidyapith, Banasthali-304022
*Institute of Advance Management and Research, Ghaziabad-201001 (UP)
Abstract
In this paper, we consider cellular radio network with integrated voice/data traffic. The blocking
probability for three channel assignment schemes (i) Non-prioritized scheme (NPS) (ii) Reserved
channel scheme (RCS) and (iii) Cutoff priority with subrating (CPS) are obtained by considering
a cluster of seven cells. In NPS, we take a simple model without giving any priority to any type of
traffic whereas in RCS we reserve some channels to serve handoff calls only. To provide efficient
service to handoff voice calls we offer subrating of channels. In subrating scheme, a new channel
in the blocked cell is created by dividing a full rate channel into two half rate channels; one to
serve existing call and the other one to serve handoff voice call. The expressions for forced
termination probability and probability of not completed calls are established. Sensitivity
analysis is provided to explore the effect of system parameters on call loss probability.
Keywords : Cellular Technology, Wireless Network, Channel Allocation, Integrated Traffic, Call
Loss Probability, Forced Termination, Subrating.
INTRODUCTION
Owing to the scarcity of the frequency spectrum,
wireless cellular technology has grown tremendously,
as the frequency channels can be used many times to
facilitate efficient utilization of limited bandwidth
available for mobile communication. Recent
advances in cellular technology have led to a
proliferation of the capability of coping with the
increasing demand of integrated traffic specially in
urban area. In the cellular systems, the geographical
area is divided into microcells, each of which
allocates a set of frequency channels in different form
allocated to neighbouring calls to avoid co-channel
interference. However, the smaller cell size in order
to increase the number of mobile users has resulted
into an undesirable consequence of an increase in
handoffs which are essential to provide continuity of
ongoing calls when mobile subscriber moves into an
adjacent cell. Performance modelling of Personal
Communication Services (PCS) Network, which
allows users to communicate while on move, has
been attempted by several researchers, in order to
improve the spectral efficiency using suitable channel
assignment scheme. As radio spectrum is a scarce
resource, the basic dimensioning issue in cellular
mobile system is to determine the number of guard
channels, assigned in each cell, so that the prescribed
level of grade of service (GOS) is achieved. As
handoff being an important function of mobility
management, several handoff strategies are
employed, based on criteria of users’ demand and
network operating costs. Some notable strategies to
take initial and handoff problems are (i) Nonprioritized scheme (NPS) (ii) Cutoff prioritized
scheme (CPS) (iii) Subrating scheme (SRS) and
(iv)Measurement based priority scheme (MPS).
These strategies can be implemented by selecting any
one of the channel assignment schemes, namely fixed
channel assignment [3,4], dynamic channel
assignment [12,13,16], directed retry [6,11],
borrowing channel assignment [5] etc. Combination
of two assignments such as fixed and dynamic
channel assignment i.e. hybrid scheme, dynamic
channel allocation [2] have resulted into better
performance. It is noted that by suitable choice of
assignment scheme, the grade of service is improved,
as blocking probability decreases to a good extent. To
reduce the blocking or forced termination of handoff
calls Lin et al. [15] proposed a subrating scheme.
According to this scheme when there is a call request
for handoff in a neighbouring cell and all the
channels in that cell are busy, then a full rate channel
splits into two half rate channels. One to serve
existing call and another to serve handoff request.
Channel subrating can be easily implemented in time
division multiplexing, such as GSM, PACS, DECT
etc. which provide protocol for subrating channel
assignment. Li and Alpha [14] gave subrating scheme
for PCS network considering Markovian Arrival
Process (MAP) with correlation between interarrival
times for modelling the system.
With the advancement of technology, integrated
services through cellular network have become
popular. Now-a-days internet services are possible
using Laptop/mobile phone through cellular network.
Earlier, work on integrated traffic was done by
Descoux [5], Atmaca et al. [1] etc. Integrated traffic
model for PCS is studied by Jain [7]. Queueing
model with mixed traffic and finite population was
also given by Jain [8]. Jain et al. [10] compared two
priority policies, CPP and TPP for mixed traffic.
Recently Jain [9] investigated prioritized channel
assignment schemes for mixed traffic with finite
buffer, which was more realistic than infinite
buffering policy.
In this paper, we consider cellular radio network with
integrated voice/data traffic. The model description
along with underlying assumptions and notations are
given in section 2. Non-prioritized scheme (NPS),
Reserved channel scheme (RCS), Cutoff priority
along with subrating (CPS) are presented in section 3.
By taking a cluster of 7 cells with arbitrary traffic
pattern, the expressions for forced termination
probability (Pft) and probability that a call is not
completed (Pnc) are computed in section 4. The
procedure to compute Pft and Pnc is also described.
Numerical illustrations to validate the analytical
results and to compare various handoff schemes are
facilitated in section 5. Finally the contribution of the
work is highlighted in section 6.
ASSUMPTIONS AND NOTATIONS
In this section, we develop a Markov chain model to
analyze the Personal Communication System (PCS).
The assumptions and notations used in defining the
channel assignment schemes are as follows :
 We consider a cellular system with cluster size
K. Each cell is allocated C channels, of which r
channels are reserved for handoff attempts in
Reserved channel scheme (RCS) and Cutoff
priority with subrating (CPS).
 Arrival process : The new and handoff request
in a cell is assumed to be originated according to
the Poisson Process with mean rate as given by
nv
arrival rate of new voice calls
nd
arrival rate of new data calls
hv
arrival rate of handoff voice calls
hd arrival rate of handoff data calls
We also denote
n= ini ; h= ihi
and n+h
where i takes value 1,2. Here suffix 1 (2) denotes
voice (data) type packets.
 Call holding and portable residence process:
o The call holding times are defined as the
duration for the call connection is requested
to a cellular network. Both new and handoff
calls, are independent and identically
exponentially distributed with rate 
o The cell residence times of each portable
defined as time interval that a portable stay
in the cell, are exponentially distributed with
rate .
 Blocking Probabilities:
o The blocking probability of new requests
denoted by BN is the probability that a new
user finds all channels busy in certain cell.
o The blocking probability of handoff voice
request (Bh1) is the probability that a handoff
voice finds all channels busy on its arrival in
the target cell.
o
o
o
The blocking probability of handoff data
packets (Bh2) is the probability that a
handoff data request finds all channels busy
on its arrival in the target cell.
A request is forced to be terminated during
its process due to a handoff failure; the
probability of forced termination is denoted
by Pft.
The probability of not completed call (Pnc) is
the probability that a call is not finished due
to blocking or forced termination.
Denote
S(k) The probability that a new attempt originated
is served by the system in the cell k.
BN(k) The blocking probability of a new attempt in
cell k.
BT
The blocking probability of new attempt in
the cluster.
ai(k) The probability that a call of type i (i=1,2)
which is being served by k cell makes a
handoff to one of its adjacent cells and
accommodated successfully.
bi(k) The handoff failure probability of attempt in
the target cell k.
q(k,l) The probability that an user in cell k departs
from the cell by the lth side (l=1,2,...,6)
Ei(k) The probability that a call of type i,(i=1,2) in
cell k is forced to be terminated after 0,1,2,...
successful handovers.
Pft(k) Forced termination probability of call which
is being serviced by cell k.
c(k) Probability that a user in cell k makes a
handoff.
Di(k) The probability that an attempt of type i
makes a handoff after zero,1,2,... successful
handoffs.
hd,i(k) The handoff departure rate of type i call
from cell k
Pn
Probability that nth call gets the service.
Pnc(k) Probability that a call, which is being
serviced by cell k is not finished due to
blocking or forced termination.
CHANNEL ASSIGNMENT
SCHEMES
In this section we describe three channel assignment
schemes as follows:
A. NON-PRIORITIZED SCHEME (NPS)
We consider a cellular system wherein each cell has
C channels serving all types of requests. We can
easily find the steady state probability of n busy
channels in the system as
n

(1)
Pn 
P0
n
n!   
C
By using the normalizing condition
P
n 0
Compute P0 by using normalizing condition as
n
 1,
Pn 
C

n
n!   
n

j
 j!   
j 0
(2)
j
When all C channels are busy, the new as well as
handoff requests are dropped. In this scheme, the
blocking probabilities of new, handoff voice and
handoff packets are same and are given by
C
BN=Bhi=Pc
(3)
C!(    ) C
 C
j

j

j!   
j 0
In this scheme, to give priority to handoff attempts,
we reserve some channels to serve only handoff
requests. This improves the performance of system
and reduces the forced termination of the handoff
calls. We reserve r channels out of C channels to
serve handoff requests only. By using the birth-death
process we calculate steady state probability P n as
(4)
1
C  r n C  r


n
 h
(5)
P0  

n
n 
n C  r 1 n!    
 n 0 n!(    )
Blocking probability of new calls can be computed as
C r
C

BN 

C
P
n C  r
n



2C
P
BN 
n C  r
(10)
n
The blocking probability of handoff data and voice
attempts are given by
B. RESERVED CHANNEL SCHEME (RCS)
n


P0 ,
0nCr

 n!   n
Pn   C  r n C  r
h

P0 ,
C  r 1 n  C
 n!   n
Using the normalizing condition, we get P0 as
1
C
n
C  r n C  r 
C  r

 h


n
n 
 n  0 n!    n C  r 1 n!    
(9)
Pn  

2C
C r r C
  hh1



n


n  C 1 n!   


New calls are blocked when all C-r channels are
busy. The blocking probability of new calls is
computed as
(6)
Blocking probability of handoff data and voice calls
is same and is given by
(7)
Bhi  PC
C. CUTOFF PRIORITY WITH SUBRATING
(CPS)
To reduce the blocking of handoff voice (type 1)
attempts, we propose the subrating of the channels.
We assume that when there is a handoff voice request
in a neighboring target cell and all channels in that
cell are busy, then the full rate channel in that cell
will be divided into two half rate channels, one to
serve existing call and other one to serve handoff
voice attempt. The steady state probabilities for this
model are determined as :
n


P ,
0nCr

n 0
 n!Cr n
C r
(8)
 h
Pn  
P0 ,
C  r 1 n  C
n
 n!Cr r n C
   h  h1
P0 ,
C  1  n  2C

n
 n!  
Now P0 is obtained by using normalizing condition as
2C
Bh 2   Pn
(11)
Bh1  P2C
(12)
n C
and
FORCED TERMINATION
PROBABILITY (Pft)
We derive general expressions for forced termination
probability (Pft) and the probability that a call is not
completed (Pnc) for a cluster of K cells having an
nonuniform traffic pattern as follows :
The probability that a new call is served by cell k is
given by
S (k ) 
(1  BN (k ))
(1  BT )
(13)
where
K
BT 
B
k 1
N
(k )
(14)
K
The probability that ith type of call served by cell k is
successfully handover to a neighbour from lth side, of
cell k is
ai (k ) 

6
 q(k , l )(1  B
 
l 1
hi
(l ))
(15)
If a ith type of call is failed to make a handoff in cell
l from cell k, then that the probability bi(k) is
obtained as
bi (k ) 

6
 q(k , l ) B
 
l 1
hi
(l )
The probability Ei(k) is calculated as
E i (k)  b i (k)  a i (k)b i (k)  a i2 (k)b i (k)
 a 3i (k)b i (k)  ................
bi (k )
(16)
1  ai (k )
After substituting the values of ai(k) and bi(k), we get
 i (k )
(17)
Ei ( k ) 
     i (k )
where

Now, in order to compute various performance
indices for an arbitrary traffic pattern, we develop the
algorithm as given below :
6
 i (k )   q (k , l )(1  Bhi (l ))
l 1
6
and
 i (k )   q(k , l ) Bhi (l )
Algorithm
l 1
So, the force termination probability of ith type of
call is given by
(1  BN (k ))
 i (k )
(18)
(1  BT )      i (k )
and the probability of not completed calls is
Pft ,i (k )  S (k ) Ei (k ) 
Pnc (k )  1 
1  B N (k )
  Bhi (k )
i 1, 2
1
(19)

=
COMPUTATION OF HANDOFF
RATES
For calculating the handoff arrival rate, we consider
hd,i(k) as handoff departure rate of ith type call from
cell k, which only depends on the arrival rate of new
calls ni. The probability that a user in cell k make a
handoff is given by
c=ci(k)=

 
(20)
Again we calculate Di(k), the probability that a call
has done 0,1,2, … previously successful handoffs as
D i (k )  c  a i (k )c  a i2 (k )c  a 3i (k )c  ...........
c

1  a i (k )
Substituting the value of ai(k), we get
(21)

(22)
     i (k )
So, the handoff departure rate of ith type from cell k,
is expressed as
 hd ,i (k )  (1  B N ) Di (k ) ni (k )
Di (k ) 

(1  BN (k ))ni (k )
Input : C, r, m, h, ni (k) (where i=1 and 2).
Output: BN, Bh1, Bh2, Pft,i, Pnc. (where i=1 and 2).
Step I : Set hd,i (k)=0.2ni(k) for i=1, 2, =1.
Step II : If ||<0.0001 for all k go to step V.
Step III : Compute P0(k) and Pn(k) according to the
scheme used Also compute BN(k) and Bhi(k) for
i=1,2.
Step IV: Compute the new values for hd,i(k) for i=1,
2 and h1, h2 using equations (23)-(24) respectively
and calculate  as
(23)
6
      q(k , l )(1  Bhi (l ))
old  hd ,i (k )  new hd ,i (k )
new hd ,i (k )
Go to step II.
Step V : Compute Pft and Pnc using equations (18)
and (19) respectively.
NUMERICAL RESULTS
For nonuniform traffic, we compute the blocking
probability of new calls (BN), handoff calls of type1
(Bh1) and handoff calls of type2 (Bh2), forced
termination probability (Pft) and probability of not
completed calls (Pnc) by using the algorithm
suggested in previous section. The cluster of seven
active hexagonal shaped cells surrounded by an outer
layer of cell for avoiding boundary effect as shown in
figure 1 is considered. Numerical results are obtained
for performance indices for three schemes namely
Non-Prioritized Scheme (NPS), Reserved Channel
Scheme (RCS) and Cutoff Priority Scheme (CPS).
For all the schemes, the number of channels for each
cell is fixed as C=10.
8
(0.78,0.82)
18
(0.42,0.55)
l 1
Thus the handoff arrival rate in neighbouring cell can
be calculated with the help of handoff departure rate
of cell k as
(24)
l 1
If the probability of leaving the cell by each side is
equal i.e. q(k,l)=1/6, then
1
hi (k , l ) 
6
6

9
(0.6.,0.87)
6
(0.42,0.82)
17
(0.12,0.35)
6
 hi (k , l )  q(k , l )  hd ,i (l )
19
(0.12,0.75)
7
(0.52,0.37)
1
2
(0.26,0.15)
5
3
(0.28,0.67)
(0.34,0.2)
16
4
(0.26,0.34)
15
(0.7,0.59)
13
(0.5,0.54)
14
(0.38,0.97)
hd ,i
11
(0.92,0.25)
(0.2,0.5)
(0.9,0.56)
10
(0.8,0.15)
(l )
l 1
Figure 1 : Cluster of Cells
12
(0.32,0.35)
8.0E-01
1.00E+01
NPS
NPS
RCS
RCS
6.0E-01
1.00E+00
CPS
Pnc
Bn
CPS
4.0E-01
1.00E-01
2.0E-01
1.00E-02
0.0E+00
1.00E-03
0.1
0.3
0.5
1
(a)
0.7
0.9
0.1
0.3
0.5
1
(b)
0.7
0.9
1.0E+00
1.00E-01
1.0E-01
1.00E-05
Bh1
NPS
1.00E-13
Bh2
1.0E-02
1.00E-09
1.0E-03
NPS
RCS
RCS
CPS
1.00E-17
1.0E-04
1.00E-21
CPS
1.0E-05
0.1
0.3
0.5
1
0.7
0.9
0.1
0.3
0.5
1
0.7
0.9
(d)
(c)
1.00E-02
NPS
1.00E+00
RCS
1.00E-06
CPS
NPS
1.00E-14
RCS
Pft2
Pft1
1.00E-02
1.00E-10
1.00E-04
CPS
1.00E-18
1.00E-22
1.00E-06
0.1
0.3
0.5
1
(e)
0.7
0.9
0.1
0.3
0.5
1
0.7
(f)
Fig. 2: Effect of 1 on (a) Blocking Probability of new calls (b) Probability of not completed calls
(c) Blocking Probability of handoff for type 1, (d) Blocking Probability of handoff for type 2 calls
(e) Force Termination Probability of type 1, (f) Force Termination Probability of type 2 calls
0.9
5.0E-01
NPS
NPS
RCS
4.0E-01
RCS
1.00E+00
CPS
CPS
Bn
Pnc
3.0E-01
2.0E-01
1.00E-02
1.0E-01
0.0E+00
1.00E-04
0.1
0.3
0.5
2
(a)
0.7
0.9
0.1
0.3
0.5
2
(b)
0.7
0.9
1.0E+00
1.00E-02
1.0E-01
1.00E-06
1.0E-02
1.00E-14
NPS
Bh2
Bh1
1.00E-10
1.0E-03
NPS
1.0E-04
RCS
RCS
1.00E-18
CPS
1.00E-22
CPS
1.0E-05
1.0E-06
1.00E-26
0.1
0.3
0.5
2
0.7
1.0E-07
0.9
0.1
0.3
(c)
0.5
2
(d)
0.7
0.9
1.00E+01
1.00E-02
NPS
1.00E-06
RCS
1.00E-01
CPS
1.00E-14
NPS
1.00E-18
RCS
CPS
1.00E-22
Pft2
Pft1
1.00E-10
1.00E-03
1.00E-05
1.00E-26
0.1
0.3
0.5
2
0.7
0.9
(e)
1.00E-07
0.1
0.3
0.5
2
0.7
0.9
(f)
Fig. 3: Effect of 2 on (a) Blocking Probability of new calls (b) Probability of not completed calls
(c) Blocking Probability of handoff for type 1, (d) Blocking Probability of handoff for type 2 calls
(e) Force Termination Probability of type 1, (f) Force Termination Probability of type 2 calls
The other parameters chosen are : the mean call
holding time  0.15, the mobile residence time 
0.02. The arrival rates of both types of new calls are
in the range 0.12-0.97 (represented as (1, 2) in each
cell below cell ID number as shown in figure 1).
1.00E+00
2.0E-01
NPS
RCS
1.5E-01
CPS
NPS
1.0E-01
Pnc
Bn
RCS
1.00E-01
CPS
5.0E-02
0.0E+00
0.01
0.03
0.05

(a)
0.07
0.09
1.00E-02
0.01
1.00E-05

(b)
0.07
1.00E-09
NPS
1.00E-13
RCS
RCS
1.0E-03
1.00E-21
1.0E-04
1.00E-25
1.0E-05
0.03
0.05

CPS
1.0E-02
CPS
1.00E-17
0.09
NPS
1.0E-01
Bh2
Bh1
0.05
1.0E+00
1.00E-01
1.00E-29
0.01
0.03
0.07
1.0E-06
0.01
0.09
0.03
(c)
0.05

0.07
0.09
(d)
1.00E-02
NPS
1.00E+00
1.00E-06
RCS
NPS
Pft1
RCS
1.00E-14
CPS
1.00E-18
CPS
1.00E-02
Pft2
1.00E-10
1.00E-04
1.00E-22
1.00E-06
1.00E-26
1.00E-30
0.01
0.03
0.05

0.07
0.09
(e)
1.00E-08
0.01
0.03
0.05

0.07
0.09
(f)
Fig. 3: Effect of  on (a) Blocking Probability of new calls (b) Probability of not completed calls
(c) Blocking Probability of handoff for type 1, (d) Blocking Probability of handoff for type 2 calls
(e) Force Termination Probability of type 1, (f) Force Termination Probability of type 2 calls
Figures 2(a)-2(f) depicts the effect of arrival rate of
type 1 calls (1) on BN, Pnc, Bh1, Bh2, Pft1 and Pft2
respectively. Figure 2(a) shows that BN for NPS is
smaller than that of RPS and CPS for the same traffic
load. Figure 2(b) indicates that the probability of not
completed calls (Pnc) is minimum for NPS. Figures
2(c) and 2(d) give the effect of 1 on blocking
probabilities of handoff calls of type 1 and type 2
calls respectively. It is observed that by providing
reserved channels to handoff calls, we can reduce the
Bh1 and Bh2. Figure 2(c) demonstrates the remarkable
effect of subrating on Bh1.
Forced termination probability displayed in figures
2(e) and 2(f) show the similar effect as in figure 2(e)
and 2(f). The curves show the increase in BN, Pnc, Bh1
and Bh2 with the increase in 1 whereas Pft1, and Pft2
show slight decrease. Figures 3(a)-3(f) depicts the
effect of 2 on BN, Pnc, Bh1, Bh2, Pft1, and Pft2
respectively. The similar trend is observed as in
figures 2(a)-2(f).
Figure 4(a)-4(f) reveals the variation in BN, Pnc, Bh1,
Bh2, Pft1, and Pft2 by varying  for three schemes,
NPS RCS and CPS. It is found that by increasing the
mobility () CPS show sharp increase whereas in
RCS and NPS, moderate increasing trend is noted for
Bh1, Bh2, Pft1 and Pft2 in figures 4(c)-4(f) respectively.
From the comparison of various graphs, we conclude
that
 The BN and Pnc are always smaller for NPS than
RCS and CPS.
 By giving priority to handoff calls there is
decrease is the blocking probability of handoff
voice (Bh1) and data (Bh2) calls, probability of
forced termination of handoff voice (Pft1) and
data (Pft2) calls but at the cost of new calls.
 Increase in (i.e. increase in mobility) increases
the blocking probabilities and probability of
forced termination of handoff calls.
2.
3.
4.
5.
6.
7.
8.
9.
10.
CONCLUSION
The channel allocation schemes based on reserved
channels and subrating policies to give priority to
handoff attempts in cellular radio system with
voice/data traffic are suggested. We have provided
expressions for forced termination probability, the
probabilities that an attempt is not completed and the
blocking probabilities. The channel allocation
schemes developed are capable to deal with nonuniform pattern of traffic. Numerical experiments
performed show the validity of the analytical results
and give the insight about the effect of variation of
various parameters on system performance. The
assignment scheme may be helpful in providing
guidelines for the design and performance prediction
of cellular communication technology. Based on
discrete
capacity
allocation
technique
the
dimensioning procedure of assigning the channels
can be easily implemented to support global roaming
in PCS.
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