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Analysis
of
a Cutoff Priority Cellular Radio System
with Finite Queueing and RenegingDropping
Chung-Ju Chang,
Senior Member,
IEEE,
Tian-Tsair Su, and Yueh-Yiing Chiang
Absrruct-
Queueing
of
new
or
handoff calls can minimize
blocking probabilities
or
increase total carried traffic. This paper
investigates a new cutoff priority cellular radio system that allows
finite
queueing
of
both new and handoff calls. We consider the
reneging
from the system of queued new calls due to caller
impatience and the
dropping
of
queued handoff calls by the system
as
they move out
of
a handoff area before being accomplished
successfully. We use signal-flow graphs and Mason’s formula
to obtain the blocking probabilities
of
new and handoff
calls
and the average waiting times. Moreover, an optimal cutoff
parameter and appropriate queue sizes for new and handoff calls
are numerically determined
so
that a proposed overall blocking
probability
is
minimized.
I.
INTRODUCTION
N
A
CELLULAR
radio system, the blocking probabilities
I
of new and handoff calls should
be
depressed as much
as possible
so
as to improve the perceived service quality
or increase the carried traffic load. In several recent papers
[5]-[7],
a system with a cutoff priority channel allocation
strategy involving queueing of new calls has been proposed to
minimize the blocking probability of handoff calls and increase
the total carried traffic; a system that provides guard channels
and a waiting queue for handoff calls to achieve a higher
probability of successful handoffs has also been studied.
In
[5],
Gukrin presented a novel approach to the study of
a multichannel cutoff priority cellular radio system, in which
the queue size for new calls is infinite and the queued calls
never renege. He considered two Poisson arrival streams with
distinct arrival rates and the same exponential service time
distributions for new and handoff calls, and obtained simple
closed-form expressions for state probabilities, in which the
signal-flow graph approach and then Mason’s formula were
utilized. This analytical method can be applied to a system
with a finite queue.
In
the case of a system with a finite queue,
however, no simple closed-form expressions for state proba-
bilities can be found; instead, algorithmic numerical methods
must
be
used to handle the corresponding computational
problem. Moreover, the computational complexity increases
with the capacity of the system buffer. In
[6],
[7],
Hong
and Rappaport described appropriate analytical models and
Manuscript received January
1994;
revised February
7,
1994; approved by
IEEWACM TRANSACTIONS
ON
NEWORKING Editor c.-L.
I.
C.-J. Chang and
Y.-Y.
Chiang
are
with
the
Department
of
Communication
Engineering and Center
for
Telecommunications Research, National Chiao
Tung University, Hsinchu, Taiwan
300,
China.
T.-T.
Su
is with the Transmission Technology Laboratory, Telecommuni-
cation Laboratories, Ministry
of
Transportation and Communication, Taiwan,
China.
IEEE
Log Number 9402107.
derived performance measures for a cellular mobile telephone
system with infinite queueing of handoflcalls; the performance
measures included blocking probability, forced termination
probability, and fraction of incomplete new calls.
Queueing of both new call and handoff calls can increase
total carried traffic as well as minimize blocking probabilities.
Therefore, as an altemative to the systems proposed in [5]
and
[6], [7],
in this paper we investigate a new cutoff priority
cellular radio system with finite queueing of both new and
handof calls. In addition, we also take into account the
reneging of queued new calls due to caller impatience
[2,4]
and the dropping of queued handoff calls as they move out of
the handoff area before being accomplished successfully
[6],
[9].
Such a cellular radio system is practical because finite
buffering is more realistic than infinite buffering and because
the related call-control packets are usually carried out on a
separate control channel [51.
Our analysis is via a two-dimensional Markov chain ap-
proach. The state probabilities can
be
obtained computationally
without any problem since the system possesses a quasi
birth-death Markovian property
[
141.
We derive blocking prob-
abilities for new and handoff calls, which are defined to contain
their corresponding reneging and dropping probabilities, via
the application of signal-$ow graphs and Mason’s formula
[l],
[8];
we also obtain average waiting times for new and
handoff calls. Moreover, we heuristically define a cost function
to investigate the optimal cutoff parameter and the suitable
queue sizes for new and handoff calls.
This paper is organized as follows. The assumptions upon
which our analysis rests are presented in Section 11. In Section
111, we derive the blocking probabilities of new and handoff
calls by using signal-flow graphs and Mason’s formula and
obtain the average waiting times for new and handoff calls. We
also provide details on how Mason’s formula is numerically
carried out in our problem and address the tractability of
the numerical computation. In Section
IV,
some numerical
examples are discussed; and overall blocking probability is
proposed as a cost function for determining an optimal priority
cutoff parameter and suitable queue sizes for new and handoff
calls. Finally, concluding remarks are given in Section
V.
11.
SYSTEM
MODEL
A
conceptual model of the new cutoff priority cellular
mobile radio system is shown in Fig.
1.
The model follows
those described in
[5],
[6],
except that it considers finite
queueing of both new and handoff calls and reneginddropping
of waiting calls. The assumptions involved in this model are
1063-6692/94$04.00
0
1994 IEEE
167
CHANG
et
al.:
CUTOFF
PRIORITY
CELLULAR
RADIO
SYSTEM
Poisson orrivol of handoff
col1
Poisson
orrival
of
new
col1
with rote
Ah
with rate
hn
the system
C
Fig.
1.
The conceptual model
of
the new cellular mobile telephone system.
stated below.
1)
The system has inputs of new and handoff calls gener-
ated according to a Poisson distribution with mean rates
of
A,,
and
Ah,
respectively.
2)
The unencumbered conversation time of a call, denoted
by
T,,
is assumed to be exponentially distributed with
mean
Up,,,.
3)
The time spent
in
a cell by the mobile associated with
a successful new (handoff) call, denoted by
Tn
(Th),
is
approximately assumed to
be
exponentially distributed
with mean
l/pn
(l/ph).
4) There are
C
channels available in the system. In order
to protect handoff calls, the system assumes that in
accessing the channels the handoff calls have priority
over the new calls and a number of channels among
C
are reserved exclusively for handoff calls. We call
the number of guard channels the
cutoff
parameter
Ch.
Thus, when a new call is originated, it can be
successfully served only if the number of idle channels
is greater than
Ch.
Otherwise, it will
be
put in the queue
or blocked due to buffer overflow. The queued new call
reneges from the queue unless it can be successfully
served within its patience time
[2],
[4]; similarly, the
queued handoff call
is
dropped from the queue by the
system as it moves out of the handoff area before being
accomplished successfully [6]. Here, the time a mobile
spends in the handoff area will be called the dwell time
of the handoff call.
5)
The system provides a finite queue with capacity
N,
for new calls during call setup and a finite queue with
capacity
Nh
for handoff calls in the handoff area.
6)
The patience (dwell) time of the waiting new (handoff)
call is denoted by
Tnq(Thq)
and is approximately
assumed to be exponentially distributed with mean
Notice that
Ah
and
pCh
are correlated with other parameters
and can
be
determined from them. Interested readers are
referred to
[7]
for details. We shall address these correla-
tions in the section below entitled Numerical Examples and
Discussion.
111.
ANALYSIS
We define (n1,nz) as the system state with probability
Pnl,nz,
where n1 is the sum of the number of occupied
channels and the number of handoff calls waiting in the queue,
n2
is the number of new calls waiting in the queue, and
0
5
n1
5
C+Nh, 0
5
n2
5
N,.
The state-transition diagram
of the system can then be obtained on the basis of assumptions
1)
through
7)
above. The diagram is shown in Fig.
2.
From this
diagram, we can obtain the state-transition equations shown
below.
(i) If n2
=
0,
then
(An
+
Ah
+
nlPch)Pnl,O
=
(An
+
Xh)Pn1-l,O
+
(121
+
l)PchPnl+l,O,
for
0
5
n1
5
C
-
ch
-
1;
l/hq(
I/Phq).
by
TH~,
is approximately assumed to be exponentially
7)
The channel holding time of a call in a cell, denoted
(An
+
cPch
+
NhPhq
+
n2Pnq)Pn~,n~
=
Ah
Pn
1-1
,n2
+
An
Pn
,
,n2-l
f
(n
1
+
1)
Pnq
Pn
1
,n&l,
for nl
=
C
+
Nh.
distributed with mean Upch.
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Fig.
2.
The
state transition diagram.
(Cpch
+
NhPhq
+
n2Pnq)pnl,nz
=
AhPnl-l,nz
+AnPnl,nz-l,for
n1
=
C
+
Nh.
(1)
The above state-transitions belong to a class of Markov
chains: the quasi-birth-death
(QBD)
process
[
141.
The structure
of the transition matrix
of
the
QBD
process is in
block
tri-diagonal form.
When a larger system is considered, the
matrix can be computationally solved by a so-called
folding
algorithm.
This type of algorithm, exploited in
[14],
begins
with a forward reduction phase and then executes
a
backward
expansion phase to find the solution. Thus, for any size system
the state probabilities can
be
obtained without any problem. In
the following, several performance measures will be derived,
including the blocking probability of
a
new call and the
probability of a call being forced into termination during
conversation.
A.
The Average Blocking Probabilities
Blocking of a new call may occur for two reasons. One
is that as a new call originates, the number of available idle
channels is less than or equal to
ch
and there are no free
buffers left in the waiting queue. The other is that although a
new call has been accepted and is waiting in the queue, it fails
to access a free channel within its patience time and
so
reneges
from the system. The reneging probability of a waiting new
call can
be
easily obtained by comparing the reneging rate with
the effective arrival rate, as in
[2], [6],
[7].
Nevertheless, we
here propose an alternative approach to obtaining the reneging
(or
blocking) probability by considering an arbitrarily selected
new call (or, say, new call of interest). We denote the blocking
probability
of
an arbitrarily selected new call by
Pl.
Pl
can
be
obtained by
C+Nh
C+Nh
N,-l
pj$'
=
pnl,N,,
+
Pnl,nzRn(nl,na)
nl=C-Ch
TZl=C-Ch
n2=O
(2)
where
R,(nl,
n2)
is the reneging probability of an arbitrarily
selected new call given that the system state is
(nl,
n2)
just
at the instant when the call is accepted and put in the waiting
queue. Clearly,
C-
ch
<
n1
<
Cf
NhandO
5
122
<
Nn
-
1.
The derivation of
R,
(n1,nz)
is more complicated than the
method used in
[2],
[6],
[7],
but it paves the way for obtaining
the average waiting times later.
We find
(1
-
Rn(n1,712))
instead of
R,(nl,n~).
(1
-
R,(nl,
722))
is the probability that the arbitrarily selected new
CHANG
et
al.:
CUTOFF
PRIORITY
CELLULAR RADIO
SYSTEM
169
call can finally get a
free
channel within its patience time,
given that the system state is (n1, nz) at the instant the call
is accepted-by the system and begins waiting in the queue.
When the arbitrarily selected waiting new call successfully
accesses a
free
channel within its dwell time, the quasi-system
state is at
(C
-
ch
-
1,0), where the quasi-system state is
defined as the system state observed by
the
arbitrarily selected
new call, excluding those waiting new calls coming after the
call of interest. In deriving
(1
-
Rn(n1,
nz)),
we use the
signal-flow graph shown in Fig.
3
to portray the transitions of
quasi-system states from the input node
yin
of state (nl,
nz)
to the output node
yOut
of state
(C
-
ch
-
1,O)
and the
respective branch gains (the probabilities of transitions). Any
intermediate quasi-system state (m1,mz) in the graph may
have three possibilities of transition: (ml
,
mz)
to (ml+l, mz),
to (ml
-
l,mz), or to (m1,mz
-
1).
The possibility of
transition from (m1,mz) to (m1,mz
+
1)
is not included.
In
this graph, the transition probability from
Yin
to (n1,
nz)
is
1
because the system state is given at
(n1,nz)
as the
arbitrarily selected waiting new call is just accepted by the
system.
Thetransitionfrom(m1,mz)
to(ml+l,mz) forC-Ch
5
ml
5
C,
0
5
mz
5
nz
5
N,
-
1
indicates the arrival of an
acceptable handoff call. We denote this transition probability
by
~ml+~,mzlml,mz.
If
C-Ch
5
ml
5
C,O
5
mz
5
nz,
the
transition occurs when the remaining interarrival time of the
handoff call, denoted by
Tt,
is smaller than the remaining
channel holding time of any of
the
ml calls in progress,
TH,,
the remaining patience time of any of the
m2
new
calls waiting in the queue, and the remaining patience time
of the accepted waiting new call of interest,
Tnq.
If
C
+
1
5
ml
5
C
+
Nh
-
1,0
5
m2
5
n2,
the transition occurs
when
Tc
is smaller than the remaining channel holding time
of any of the
C
calls in progress,
TH,,
the remaining dwell
time of any of the (ml
-
C)
handoff calls waiting in the
queue,
Thq,
the remaining patience time of any of the
m2
new calls waiting in the queue, and the remaining patience
time of the accepted waiting new call of interest,
Tnq.
Since
Te , TH, Tnq,
and
Thq
are mutually independent and are all
assumed to be exponentially distributed,
Pml+l,mz~ml,mz
can
We denote the probability
of
transition from (m1,mz) to
(mi
-
1,md by
Pm1-l,mzlml,m2
for mi
=
C
-
Ch,mz
=
If ml
=
C
-
Ch,mZ
=
0
or
C
-
ch
+
1
5
ml
5
C,O
5
mz
5
nz,Pml--l,mzlml,mz is contributed by the probability
that the ml channels in use are reduced by
1
due to completion
of a conversation. If
C
+
1
5
ml
5
C
+
Nh,O
5
m2
5
nz,
Pml-l,mzlml,mz
is contributed by the probability that (i)
the
C
channels in use are reduced by
1
due to completion
of a conversation of (ii) the number of handoff calls waiting
in the queue is reduced by
1
due to dropping of a call. In
a manner similar to that used to derive
Pml+l,m21ml,m2
in
(9,
Pml-l,mzlml,mz
can
be
obtained by
(4)
shown at the
bottom of the page,where
T&
is the channel holding time
of the other (ml
-
1)
calls in progress and
TAq
is the dwell
time of the other waiting handoff calls.
T&
has the same
distribution as
TH,,
and
TAq
has the same distribution as
We denote the probability of transition from (ml, mz) to
(m1,mz
-
1)
by
Pml,mz-11ml,m2
for
C
-
ch
5
ml
5
C
+
Nhl
1
5
mz
5
n2
5
N,
-
1.
If ml
=
C
-
Chl
1
5
m2
5
n~,P~~,~~-1l~~,~~ is contributed by the probability
that (i) a call among ml now in progress will complete its
conversation or (ii) there is a waiting new call among
m2
reneging from the system. If
C
-
ch
+
1
5
ml
5
C
+
Nh
1
5
m2
I
nz,Pml,mz-llml,mz is contributed by the probability
that there is a waiting new call reneging from the system.
Accordingly,
Pml,m2-llml,mz
is given by (see
(5)
shown
below), where
TAq,
the patience time
of
the other waiting new
calls, has the same distribution as
Tnq.
Based on the established signal-flow graph shown in Fig.
3
and the branch gains obtained in
(3)-(5),
(1
-
R,(nl,nz)),
the probability that a new call attempt will succeed, can be
0
OT
c-
Ch
15
mi
5
c+Nh10
5
mz
5
nz
5
N,
-
1.
Thq.
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Fig.
3.
The
signal-flow graph
for
obtaining
Rn
(n
1.
712
).
obtained by using the general gain formula (Mason's rule)
[8],
which is given by
k=l
where
N
is the total number of
forward
paths,
which are
defined to
be
paths from the input node
(nl,
n2)
to the output
node
(C
-
Ch
-
1,0),
Mk
is the k-th forward path gain, which
is the product of the branch gains encountered in traveling
the k-th forward path,
A
=
1-
(sum of the gains of all
individual loops)
+
(sum of products of gains of all possible
combinations of two nontouching loops)
-
(sum of products
of gains of all possible combinations of three non-touching
loops)
+...,
and
Ak
=
the
A
for that part of the signal-flow
graph that is nontouching with the k-th forward path. Note that
loops are called
nontouching
if they do not share a common
node.
For the signal-flow graph shown in Fig.
3,
we find that
the graph excluding the output node
(C
-
Ch
-
1,0)
has a
rectangular structure with
(n2
+
1)
rows and
(Ch
+
Nh
+
1)
cblumns. It has
(Ch
+
Nh
+
l)n2
forward paths and
(Ch
+
Nh)
x
(n2
+
1)
individual loops. Loops belonging to
different rows or belonging to the same row but not adjacent
to each other are non-touching. The gain of each loop is
simply the product of the gains of two branches. On the
basis of the specific features of the signal-flow graph, we can
numerically compute
A
and
Ak
using a recursive algorithm.
However, the large number of
(ch
+
Nh
+
forward paths
prevents our analytical method from applying to all cases.
Fortunately,
Nn
and
Nh
need not be large in real applications,
due to call reneging and dropping. We shall examine this
characteristic in the numerical examples discussed in the next
section. In summary, the analytical method presented here is
computationally tractable. Via the general gain formula in
(6),
we can numerically obtain
R,
(nl
,
n2)
and in
turn
the blocking
probability of an arbitrarily selected new call
PF
in
(2).
We also derive the blocking probability of a handoff call
by considering an arbitrarily selected handoff call (or, say,
handoff call of interest). Blocking
of
an arbitrarily selected
handoff call occurs in two situations. The first is that there
are no free channels and no free buffers available as the call
moves into a handoff area. The second is that, although the
handoff call has been accepted by the system and is waiting in
the queue, the call cannot access a free channel within its dwell
time in the handoff area and
so
is dropped from the queue by
the system. The blocking probability
of
the arbitrarily selected
handoff call, denoted by
Pf,
can
be
similarly obtained by
C+Nh-1
Nn
NTl
PB"
=
PC+Nh,n2
+
Pnl,lzzRh(nl,nz)
(7)
nz=O
nl=C
nz=O
where
Rh
(n1,
n~)
is the dropping probability of the arbitrarily
selected handoff call given that the system state is
(n1,nz)
just at the instant when the call is accepted by the system and
waits in the queue.
In this case, we again find the probability
(1
-
Rh(n1,nz))instead of
Rh(n1,nz).
In
obtaining
(1
-
Rh(n1,nZ))
for
C
5
n1
5
C
+
Nh
-
1,
O
5
712
5
N,,
we once again construct a signal-flow graph and find the
respective branch gain. When the arbitrarily selected waiting
handoff call successfully accesses a free channel within its
dwell time, the quasi-system state is at
(C
-
l,nz),
where
the quasi-system state is defined as the system state observed
by the arbitrarily selected waiting handoff call, excluding
the new and handoff calls coming after the call of interest.
Fig.
4
shows a signal-flow graph that portrays the transitions
of
quasi-system states from the input node
yin
of state