MAP 1 , MAP 2 /M/c retrial queue with guard channels and its application to cellular networks

Sociedad de Estadlstica e Investigaci6n Operativa Top (1999) Vol. 7, No. 2, pp. 231-248

MAP1, MAP2/M/c retrial queue with guard channels and its application to cellular networks

B o n g D a e C h o i Department of Mathematics, Korea University 1, Anam-dong, Sungbuk-ku, Seoul 136-701, Korea e-mail : [email protected]

Y o n g C h a n g Information and Telecommunication Business Samsung Electronics Co., LTD.,263 Seohyeon-Dong Pundang-Gu, Sungnam-Shi, Kyunggi-Do, 463-050, Korea e-mail : [email protected]

B a r a K i m Department of Mathematics and Center for Applied Mathematics, KAIST, 373-1 Kusung-Dong, Yusong-Gu, Taejon, 305-701,Korea e-mail : [email protected]

A b s t r a c t

In this paper, we investigate the impact of retrial phenomenon on loss probabilities and compare loss probabilities of several channel allocation schemes giving higher priority to hand-off calls in the cellular mobile wireless network. In general, two channel allocation schemes giving higher priority to hand-off calls are known; one is the scheme with the guard channels for hand-off calls and the other is the scheme with the priority queue for hand-off calls. For mathematical unified model for both schemes, we consider the MAP1, MAP2/M/c/b, c~ retrial queue with infinite retrial group, geometric loss, guard channels and finite priority queue for hand-off calls. We approximate the joint distribution of two queue lengths by Neuts' method and also obtain waiting time distribution for hand-off calls. From these results, we obtain the loss probabilities, the mean waiting time and the mean queue lengths. We give numerical examples to show the impact of the repeated attempt and to compare loss probabilities of channel allocation schemes.

Key Words: Retrial Queue, MAP, Channel Allocation Scheme, Matrix-geometric Method A M S subject classification: 60K25, 60K30, 90B12, 90B22

1 Introduct ion

The study of the queueing system presented in this paper has application to traffic performance in cellular mobile networks. Before stating our model,

Received: September 22, 1998; Accepted: February 12, 1999

232 B.D. Choi, Y. Chang and B. Kim

we will provide some information on cellular mobile network. In the cellular mobile wireless network, the regional service area is divided into a certain number of cells, each of them served by a base station(BS) having a limited number of channels. A mobile user (also called mobile station) has wireless terminal such as handy-phone for voice service, wireless data terminal for data service, dual terminal for voice and data, and so on.

We consider one specific cell in the cellular mobile networks, which faces two kinds of call arrival process; originating (or fresh) calls, i.e., calls originating in that cell, and hand-off calls, i.e., current ongoing calls caused by mobile users moving from one cell to that cell. The fresh call originating in that cell requests an idle channel (frequency, time slot, code) to the BS for a call setup. If there are one or more idle channels in the BS of that cell, then the BS assigns an idle channel to the mobile user and the call is connected until the conversation is completed or the mobile user moves out of that cell (and a hand-off occurs). When the current ongoing call is moving from adjacent cell into that cell, the channel in the old cell is released and a channel is required in the new cell, and mobile user sends the request message for acquiring an idle channel to the BS of the entering cell. If there are one or more idle channels in the BS of the that cell, the call is successfully handed over without a service interruption; otherwise he waits in the finite queue or he is forced to terminate in the case of full queue before call completion. The current ongoing call moving from one cell to another cell which requests an idle channel to the entering cell is called the hand-off call.

The degradation of quality of service (QoS) caused by such a breakdown of hand-off calls is more serious than that caused by a blocking of an originating call. In order to reduce the probability of the forced termination, We need to give higher priority to the hand-off call over the originating call. One of them is the scheme with guard channels with no retrial proposed by Gu~rin (1988), and Yoon and Un (1993). These guard channels can only be used by hand-off calls. Another scheme is to provide a priority queue only for hand-off calls proposed by Hong and Rappaport (1986) with no retrials, and Choi and Chang (1999) with retrials. When hand-off call finds no idle channels; then he waits and use the current channel for a while (which means hand-off call enters to priority queue) and then, as soon as channel is available, he acquires one channel.

The blocked originating call may retry for service after a short period.

Retrial queue with guard channels 233

We name such a blocked call a retrial call(or repeating call). After several reattempts, either the call is connected or the mobile user decides to give up the call attempt. The blocking of originating call occurs more often in the wireless networks than in the wired networks because wireless frequencies as physical channels are limited. With modern mobile handsets, redialing of blocked calls is only a matter of pushing one button. So the time spent by blocked users before starting a retrial becomes shorter compared to the conventional telephone systems. These retrial calls will give a negative influence on hand-off call connections as the offered load of the system becomes higher. This is another reason why we give higher priority to hand- off calls. Therefore we need to investigate the impact of retrial phenomenon and the performance of channel allocation schemes.

There is a large number of papers dealing with retrial phenomenon; most of them are retrial queues with single type of calls (see Falin and Tem- pleton (1997) and Yang and Templeton (1987)). For a complete survey on retrial queues with two types of calls, see Choi and Chang (1999). For retrial queues with two types of calls which emphasize the applications to cellular mobile networks, see Carothers and Lin (1995), Tran-Gia and Mandjes (1997), Choi, Choi and Lee (1995), and Choi and Chang (1999). Carothers and Lin (1995) studied M1, M2/M/c/c, co retrial queue with no priority for hand-off call by simulation method. Tran-Gia and Mandjes (1997) studied M1,M/M2/c/c,c~ retrial queue to investigate retrial phenomenon in cellular mobile networks by the analytical method. They also investigated M/M/c/c , co retrial queue with guard channels to prioritize the hand-off call. Choi and Chang (1999) considered MAP1,MAP2/M/c/co, k retrial queue with priority queue for hand-off calls and obtained joint distribution of two queue lengths by Neuts' matrix-geometric method, and gave numerical example to show the effect of retrial phenomenon and the channel allocation scheme with priority queue for hand-off calls.

An important issue for network planning is the modeling of hand-off call process. The following is one of reasons why we model hand-off arrivals as MMPP(Markov Modulated Poisson Process) (here we assume MAP(Markov Arrival Process)). In the recent wireless networks such as UMTS(Universal Mobile Telecommunication System), IMT(International Mobile Telecommunication)-2000, the geographical cell structure is com- posed of the pico-cell, micro-ceil and macro-cell, i.e., multi-layered cells. When the originating call arrives at the BS for acquiring idle channel, if there are no idle channels in the BS of the pico-cell, the BS of the micro-cell


which connects several pico-cells assigns an idle channel to the originating call which is overflowed from the pico-cell. The overflowed call arrival process from the subordinated cells generally is modeled as MMPP. We consider MAP which generalizes MMPP.

In this paper, we investigate the impact of retrial calls and the performance of channel allocation schemes giving the higher priority to hand-off calls. Two channel allocation schemes giving priority for hand-off calls are known; one is the scheme with the guard channels for hand-off calls and the other is the scheme with the priority queue for hand-off calls.

To obtain unified results for both systems, we consider the MAP1, MAP2/M/c/b, oo retrial queue with infinite retrial group, geometric loss, guard channels and finite priority queue for hand-off calls. Above two schemes are special cases of our model.

This paper is organized as follows: In section 2, the mathematical queueing model is described in detail. In section 3, by Neuts' method we approximate the equilibrium joint distribution of two queue lengths and the loss probabilities of hand-off calls and originating calls. In section 4, we derive the waiting t ime distribution of hand-off calls. In section 5, we give numerical results to show the impact of retrial calls and the performance of channel allocation schemes giving the higher priority to hand-off calls.

2 M o d e l D e s c r i p t i o n

In this section, we give mathematical model of one cell with the channel allocation schemes with both guard channels and priority queue for hand- off calls. The cell has two types of calls: originating calls and hand-off calls. We model this system by the multiple server retrial queueing system with both priority queue and guard channels(see Figure 1). The hand-off calls arrive to that cell according to MAP1 with a representation (C(1), D(1)) and underlying Markov chain X1 (t) with the dl x J1 generator C(1) + D(1), and the 1 x J1 stationary vector 6(1). The originating calls arrive to that cell according to MAP2 with a representation (C(2), D(2)) and underlying Markov chain X2(t) with the J2 x J2 generator C ( 2 ) + D ( 2 ) , and the 1 • J2 stationary vector (I(2). MAP1 and MAP2 are independent each other. The fundamental arrival rates of MAP1 and MAP2 are A1 - (f(1)D(1)ejl and A2 --- (f(2)D(2)ej 2 respectively. There are c(c >_ 1) identical independent

Retr ia l queue with guard channels 235

Hand-o f f cal l MAP~

Or ig inat ing cal l MAP 2

f (.

c -channe ls

~ a r d ~'~kCell d e p a r t u r e ' - - J channels

, " . . . . . i

: i ~ i Serv ice [ Depa r tu re

i _ _ _ _ >

i i ( / l - q ( l - p : 1 -s t t i m e )

/ / " ~q{{p : l - s ~ ~- Geomet r , c ~ - ~ Loss

Retrial G roup ~ ,

Q u e u e wi th f inite c a p a c i t y

F i g u r e 1: The Model Description

channels with Ch(C h __> 0) guard channels for hand-off calls. The service times of hand-off call and originating call are independent and identically distributed with the exponential distribution with parameter #. The cell- sojourn times of any ongoing calls(he is on conversation) in that cell are independent and identically distributed with the exponential distribution with parameter v. On the arrival of a hand-off call to that cell, if all channels are occupied, the hand-off call enters the finite priority queue with the capacity b - c if there exist free waiting positions. If all waiting positions in the priority queue are occupied, then the hand-off call is lost and so the mobile user during a conversation is forced to terminated. On the arrival of an originating call to that cell, if the number of channels more than c - C h are occupied, the originating call either leaves the system forever with probability 1 - p or leaves the service facility temporarily with probability p to retry for service after a random period of time. The retrial time is defined as the amount of time between two consecutive attempts made by a retrial call and is independent of all previous retrial times. The retrial times are distributed with the exponential distribution with parameter 3'. On the retrial, if retrial call finds that the number of channels more than c - Ch are still occupied, then he returns to the retrial group as a retrial call with probability q or he leaves the system forever with probability 1 - q. The capacity of the retrial group is infinite. If mobile user cannot acquire an idle channel in a limit amount of time during staying in the handover region, a hand-off call in priority queue may leave without connection after exponential random variation with parameter V 1 (121 > b').


3 T h e M a t r i x - G e o m e t r i c S o l u t i o n

Let Nl(t) be the number of retrial calls in the retrial group at t ime t, N2 (t) be the number of calls in the priority queue including busy channels at t ime t, and X1 ( t ) ,X2(t) be the states of underlying Markov chain for MAP1 and MAP2 at time t respectively Then (N1 (t), N2(t), X1 (t),)(2 (t)) is Markov process whose state space is

,-q =-- {(i ,n, j l , j2 ) : 0 ~_ i ,0 < n < b, 1 ~ j l ~ J l , 1 _~j2 _~ J2}.

We order the elements of S lexicographically. Then the infinitesimal generator Q of Markov process (Nl(t) , N2( t ) ,X l ( t ) ,X2( t ) ) has QBD type as follows.

Q =

B1 Ao 0 0 . . . 0 --. B2 B I - F Ao 0 .-- 0 --- 0 2B2 B1 - 2F Ao .-. 0 ---

�9 . �9 �9 . -

�9 �9 �9 . �9 �9

0 0 . . . nB2 B I - n F Ao "" . . . . �9 �9

�9 , , * .

The entries of Q are given by the following block matrices�9

Ao = D i a g ( 0 , - - - , 0 , / J 1 @ p D ( 2 ) , " "

(c - ch) -th

, I& @vD(2) )

F = Diag(qlJ1J2," �9 �9 , 7Ij t J2, (1 - q)TIjt J ~ , " "

(c - Ch)-th

0 '~IJ1 J2 0 . . . . . . . . .

* . ".. . . . . . . .

0 "7[J1 J2 0 �9 �9 �9

B 2 = 0 ( 1 - - - q ) ' Y l J i J 2 0 . . .

",. ".. " .

. . . 0 ( 1 - . q ) q l j t j 2

. . . . . . . . . 0

, (1 - q)71j~j2)

0

( l q ) q l J l J2

O-th

(c - ch - l)-st

(c -- Ch)-th

(b - 1)-st b-th


S l =

Sl T1 0 . . . . . . . . . . . . . . . . . . U l - U I + S 1 T1 0 . . . . . . . . . . . . . . .

�9 . . . �9

" ' " U c - c h - U c - c h + S 2 T2 0 . . . . . . . . .

�9 . . . .

�9 .

. . . . . . ". Uc - U c + S 2 T2 0 . . .

. . . . . . . . . . . . U c + I - U c + I + S 2 T 2 - . .

. . . . . . . . . . . . . . . . . . V b - U b W S 2 + T 2

where

and

$1 -= C(1) @ C(2), T1 - D(1) q~ D(2),

$2 = C(1) ~ (C(2) + (1 - p)D(2)) , T2 =- D(1) | Ij2,

n ( # + v ) I n = O , . . . , c , U n - c ( # + v ) + ( n - c ) v l I n = c + l , . . . , b .

The generator Q is the level dependent quasi bir th-and-death matrix. Let x be the steady state probability vector, which is the unique solution of linear equations x Q = 0, xeoo = 1; where eer is an infinite dimensional column vector whose elements are all equal to 1. The parti t ion of x is denoted by x -- (x (0 ) ,x (1 ) , - . . ) and the parti t ion of x(i) is denoted by 'x( / ) = (x(i ,0) , -- . ,x(i ,b)). The component x( i ,n) of x(i) is the 1 • JIJ2 vector such that x( i ,n) = (x(i,n, 1, 1),--- ,x( i ,n , j y , j 2 ) , ' " , z ( i ,n , J1, J2)), where i > 0,0 < n < b. x(i ,n, j l , j2) is the probability that Markov process is in state (i, n, j l , j2) in the steady state.

It is still an open problem to express x in the closed form (e.g., gener- ating function). So we use Neuts' approximation method (Neuts and Rao (1990)). We assume that only k calls among the retrial calls in the retrial group can retry for the service even if there are retrial calls greater than k in the current retrial group. A choice of appropriate value of k is done by a method in Neuts and Rao (1990). Then the infinitesimal generator Q*


of the approximated system is given by

Q . =

B1 A0 B2 B1-F 0 2B2 :

0 0 0 0 0 0 :

0 0 . . . 0 0 0 . . .

Ao 0 . . . 0 0 0 . . . B1-2F A0 --- 0 0 0 --.

�9 . . �9 �9 �9

. �9 �9 . . o

�9 .. kB2 B I - k F Ao . . . 0 . . .

�9 .. 0 A2 A1 Ao 0 . . . �9 . . 0 0 A2 A t Ao "'"

�9 �9 o �9 . �9

�9 . . o . . .

A2 - kB2, A1 = B1 -- kF

Let A = A0 + A1 + A2, then we easily see by the s t ructure of model that A .is an irreducible infinitesimal generator matr ix and so, there exists 1 • (b + 1),11,12 stat ionary probabili ty vector zr of A,i.e, ~rA = 0,1re -- 1, where e - - e ( b + l ) j 1 j 2 , en, n > 1 is n • 1 column vector whose every elements are equal to 1. The vector 7r is denoted by ,r = (,r(0), ,r(1), �9 �9 �9 , ,r(b)) whose components are lr(n) = (Tr(n, 1, 1 ) , . . . , 7r(n, j l , j 2 ) , ' . . , 7r(n, J1, J2)), for 0 <_ n < b. r ( l , j l , j 2 ) is the s ta t ionary probability that there are l calls in the queue including busy channels and the underlying Markov processes of the MAP( l ) , MAP(2) are in j l and j2, respectively, for modified system so as to append a retrial call to the retrial group immediately when the retrial group size goes to below k.

T h e o r e m 3.1. The Markov process with the generator Q* is positive recurrent i f and only i f

b

7 r ( n ) ( I j l | < 7k (3.1) n _ ' : C - - C h

P r o o f . Since the Markov process has the QBD structure, Markov process is positive recurrent if and only if (Neuts (1981))

zr Aoe < ~r A2e,

By subst i tut ing A0 = D i a g ( 0 , . . . ,0,/d~ | pD(2 ) , . - . , / j , | pD(2)) and A2 = kB2,

b e - c h - 1 b

P Z ~ ( n ) ( I j ~ @ D ( 2 ) ) e < 7k Z r ( n ) e + ( 1 - q ) T k Z ~(n)e . n = C - - C h n ~ - - - 0 n - - - - C - - C A


Thus (3.1) follows. []

R e m a r k 3.1. Intuitive interpretation of (3.1); Since LHS of (3.1) is rate into retrial group and RHS of (3.1) is maximum rate out of retrial group, (3.1) should be necessary and sufficient condition for positive recurrence.

Under the stability condition of this system in Theorem 1,we can get the following results. For the simplicity, we define

H r n - ( - m B 2 ) [ H m - l A o + B l - ( m - 1 ) r ] -1 m = l , 2 . . - , k

where H0 = 0.

T h e o r e m 3.2. The unique steady-state probability vector x satisfying xQ* = 0, xec~ = 1 are given by

1-Im=i+lHm, for 0 < i < k - 1 , x ( i ) = x(k) k x(k)R i-k, for i > k.

The vector x(k) is obtained by solving

x(k)[HkAo + A1 + RA2] = O,

with the normalizing condition

k - 1 k

x(k)['~--~ H g m + ( I - R)- l ] e = 1 i=O r n = i + l

where R is the minimal nonnegative matrix solution to the matrix-quadratic equation R2A2 + RA1 + Ao = O.

P r o o f . The matrix equation xQ* = 0 is as follows.

x(0)Sl + x(1)B2 = 0

x(0)Ao + x(1)(B1 - F) + x(2)(2B2) = 0

x(k - 2)A0 + x(k - 1)(Hi - - ( k - 1)F) + x(k)A2 = 0

x ( k - 1)Ao +x(k)A1 + x ( k + 1)A2 = 0

x(k)Ao + x(k + 1)A1 + x(k + 2)A2 = 0

For n > k, x(n)Ao + x(n + 1)A1 + x(n + 2)A2 = 0


We obtain from the above system equations that x ( m - 1) = x(m)Hm, m - - 1,2,. .- ,k. Thus,

x(i) = x ( k ) I I Hm, r n = i + l

0 < i < k - 1 .

Since lrA2e > lrAoe, there exists rate matrix R such that x(k+i) -- x(k)R i, i > 0, where the maximal eigenvalue of R is less than 1, i.e, sp(R) < 1 and R is the minimal nonnegative matrix solution to the matrix-quadratic equation R2A2 + RA1 + Ao = 0 (Neuts (1981)). From the equation x(k - 1)Ao + x(m)A1 + x(m + 1)A2 = 0 and x(m + 1) = x(m)R, we have

x(k)[HkAo + A1 + RA2] = O. (3.2)

Thus x(k) is a solution of (3.2) with the normalizing condition,

x,k i R, 1]e=1 Li=O m----i+l

[]

The matrix R is approximated by the following iteration

R(0) = 0

R ( n + l ) = - A 0 . A 1 1 -

Iterations will be continued until

R2(n) " A2 " A~ 1 for n > 0

max/,/ [R/j (n + 1) - R/j(n)] < e

is satisfied, where R(n) is n-th iteration and e is the degree of accuracy required. The accuracy can be checked by RA2e = Aoe. ( Neuts (1981), p.14 theorem 1.2.3).

Performance Measures

We will find the loss probability of hand-off call and originating call. Let OO

y _-- x(i) (3.3) i =0

I . i=0 m = i + l


then y is the 1 • (b+l )J i J2 probability vector such that y = (y(0), y(1),- �9 �9 , y(b)) and y(n) = (y(n , l ,1) , . . . ,y(n, J1,J2)) for 0 < n < b. y(n, jl,j2) is the probability that there are n calls in the priority queue including busy channels and the underlying processes MAP(l) and MAP(2) are in j l and j2, respectively, in steady state.

We denote PL1 for the loss probability of hand-off call and PL2 for the loss probability of originating call. PL1 and PL2 is given by

1( b ) PLI=-~I Y(b)(D(1)| + ~ Y(n)egl j2(n-c)vl '

n = c + l

(3.4)

PL2 = ~22 ( 1 - p ) y ( n ) ( / j 1 @ D(2))ej1j2+ n:C--Ch

} +~-'~ E i ') '(1-q)x(i'n)ej1j2 " i----0 n=c--ch

(3.5)

The mean number of retrial calls in the retrial group E[N1] is given by

E[N ] oo

= i.x(i)e i = 0

k

--- ~ i . x(i)e + x ( k ) [ k R ( I - R) -1 + R ( I - R ) - 2 ] - e . (3.6) i=O

The mean number of calls in the priority queue including busy channels E[N2] is given by

b

E[N2] = ~ n . y(n)ej1j 2 n----0

b cr

= Z ) - ~ n . x ( i , n ) . e j 1 j 2 (3.7) n = O i = 0


4 W a i t i n g T i m e D i s t r i b u t i o n

In this section, we derive the waiting time distribution of hand-off call. Let W be the random variable representing the waiting time of a hand-off call who is not lost. The waiting time is the time period during which the hand-off call waits in the priority queue until he acquires an idle channel. Let W*(s) be the LST(Laplace-Stieltjes transform) of W. Conditioning on N2(t) immediately before arrival epoch of a tagged hand-off call who will acquire idle channel, we have

b-1

W*(s) = ~ anW~(s), (4.1) n=O

where an is the probability that there are n calls in priority queue at arrival epoch of a tagged hand-off call who will acquire an idle channel, and W* (s) is the LST of waiting time of a tagged hand-off call given that there are n calls in priority queue immediately before arrival epoch of the tagged hand-off call and he will acquire idle channel. First, an is given by the ratio of the rate of hand-off call arrivals who find n calls in the priority queue including busy channels and will acquire an idle channel and the rate of hand-off call arrivals who will acquire an idle channel, that is,

y(n)(D(1) | Ij2)eJ, j2rn (4.2) an = ( 1 - P L 1 ) A a '

where rn is the probability that a hand-off call acquire an idle channel given that N2(t) -- n immediately before the arrival epoch of the hand-off call, and is given by

1 if n < c, rn = c(u+~) (4.3)

c(u+~)+(n-~+U~ if n > c.

Using the memoryless property of exponential distributions, W~ (s) is given by

i fn<c , (4.4) Wn(8 ) -- 1 c ( ~ -~- v ) -~- Ip l i f n > c.

l=l 8 A- C(l~ -4- u) A- lUl


By substi tuting (4.2), (4.3) and (4.4) into (4.1), we have

1 c--1

W*(s) = (i- PLI))~I {n~__o y(n)(D(1) | Ij2)eJIJ2

b-1 C(~ -'b 12) + ~-:~Y(n)(D(1)| 1)u 1

n-c+l c(# -4- u) -4- llJl } II s+ccu+ )+l.1 "

l----1

The expected waiting t ime E[W] is given by

E[W] - dsW*(s) [s=0

b - 1

= 1 �9 ~--~ y (n ) (D(1 ) | Ij2)ejxJ2 ( I - PLI)AI ,~=c

c(p + u) n-c+1 1 X c ( # + u ) + ( n - c + l)ul ~ c(#"bv)+lVl" (4.5)

Performance measure E[W] will be used when we want to decide the capacity of priority queue in order to have E[W] less than specific value.

5 N u m e r i c a l E x a m p l e s

In this section, we present numerical results to investigate the performance evaluation of channel allocation schemes with and without retrials in the cellular mobile networks. We consider three priority disciplines of providing priority to hand-off calls as follows: Scheme I is only to give guard channels for hand-off calls where this is the case of b - c = 0 in our mathemat ical model, Scheme II is to give priority queue for hand-off calls where this is the case of ch = 0 in our mathemat ica l model and Scheme III is to give both guard channels and priority queue to hand-off calls.

Now, we compare loss probabilities PL1 and PL2 as performance measures for three channel allocation schemes with and without retrials.

In the following numerical examples, parameters are set as the follow- ings: k = 30, q , = 4, u = 1, ul = 3, /~ = 1, p = 0 . 9 , q : 0 . 9 a n d c = 10.


1

0.1

001

~ 0.001

~1 ~06

1e-07

1 e-06

le-09

With Retrial Ch=O " Without Retrial - -

h=2

Ch=4

2 4 6 8 10 12 14 16 18 20

Total Arrival Rate

F i g u r e 2: Loss Probability of Hand-off calls PL1.()~1/~2 = 0.3 : Scheme I)

For simplicity we assume that hand-off calls and originating calls arrive to the system according to Poisson processes with ~1 and ),2 respectively. We assume the case ~1/),2 = 0.3, which means that the hand-off calls occur 0.3 times than the originating calls occur.

First, we check the effect of retrial phenomenon on loss probabilities. In Figures 2-5, we see that loss probabilities of hand-off (originating) calls are larger (smaller) in the model with retrials than in the model without retrials. We also see that retrial calls give strong effect on loss probabilities of both calls with heavy offered load. Finally, Figures 2-5 show that small change in the number of guard channels (and buffer size) produces big dif- ference in loss probabilities of both calls.

Next we compare the performance of three schemes with retrials in terms of loss probabilities. Figure 6-7 display PL1 and P L2 for three schemes versus the total arrival rate of calls with )`1/)`2 = 0.3. Among three schemes Scheme II gives the lowest loss probability of hand-off calls at the expense of the highest loss probability of originating calls.


1 i i i i i

0.1 With Retrial Without R e t r i a l - - ~ ~ ........

0.01 ..... m

(~0.001

~ . 0 0 0 1 o

~ 1e-05

= o, le-06

I a-07

1 e-08 i I i i 0 2 4 6 8 10 12 14 16 18 20

Total Arrival Rate

F i g u r e 3: Loss Probability of Originating calls PL2.(11/~2 = 0.3 : Scheme I)

i i i

0.1 With Retrial .... " ~ Wdhout Retrial - -

0.01

~ 0.001

~ . 1 ~ 0 1

~ 1 *4~5 ~

ft.

le-06

I e-07

I e - 0 8 i 2 4 6 8 10 12 14 16 18 20

Total An'ival Rate

F i g u r e 4: Loss Probability of Hand-off calls PLI.(A1/A2 = 0.3 : Scheme II)


0.1

0.01

0.001

(b ._~.0001

le-07 I

le-08

Ie-09

le-10

, , [ i , , , , i

With Retrial ~ "

B ~ - O Bz]ffer=2

/ B

. . . . . , '8 2 4 8 10 12 I I I

Total Arrival Rate

20

F i g u r e 5: Loss Probability of Originating calls PL2.(A1/A2 = 0.3 : Scheme II)

l

0.1 Buffer=O, Ch-4 - - Buffer=4, Ch~l . . . . . . . . . . . . . . . . . . . . . . . -" Buffer=2, Ch=2 . . . . . . . . . . . . . . . . . .

0.01 . . . . . . . .

O.OOt

~ . ~ 1

o

~ le-07

3 le4~

1e-10 . . . . 0 ' 4 ' 8 2 4 6 8 I 12 I 16 I 20

Total Arrival Rate

F i g u r e 6: Loss Probability of Hand-off calls PLI.(•I/A2 = 0 . 3 )


0.1

0.01

~ 0.001

o l e -05

..a llM17

l e 4 ~

Buffee-O, C1"~4 - - e~ ,~ .4 , c , - o ..... __.---------- :: ::-:::: ....... Buf fer .2 , C h - 2 . . . . ~ . . . . . . . . .

2 4 6 8 10 1 14 1 1 20

Total Arrival Rate

F i g u r e 7: Loss Probability of Originating calls PL2.(A1/A2 = 0.3)

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MAP 1 , MAP 2 /M/c retrial queue with guard channels and its application to cellular networks

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