[8], Atencia and Moreno [9-10], Moreno [25], Wang and. Zhao [28] ... discuss an M/M/1 retrial queue with working vacations and Li et al. [23] gives a. 1. Geo Geo.
Vol. 10, No. 4, pp. 495-512, 2013
© ICAQM 2013
Discrete-Time //1 Retrial Queue with General Retrial Times, Working Vacations and Vacation Interruption Shan Gao1, 2 and Jinting Wang1 2
1 Department of Mathematics, Beijing Jiaotong University, Beijing, China Department of Mathematics, Fuyang Normal College, Fuyang, Anhui, China
(Received November 2012, accepted March 2013)
______________________________________________________________________ Abstract: We consider a discrete-time Geo X G 1 retrial queue with general retrial times, and introduce working vacations and vacation interruption policy into the retrial queue. Firstly, we analyze the stationary condition for the embedded Markov chain at the departure epochs. Secondly, using supplementary variable method, we obtain the stationary probability distribution and some performance measures. Furthermore, we prove the conditional stochastic decomposition for the queue length in the orbit. Finally, some numerical examples are presented. Keywords: Batch arrival, discrete-time queue, retrial, vacation interruption, working vacation.
______________________________________________________________________ 1. Introduction
R
etrial queues are characterized by the feature that arriving customers who find the server unavailable will join the retrial group to try their luck again some time later. During the last two decades, retrial queues have been investigated extensively because of their applications in telephone switching systems, telecommunication networks and computer systems for competing to gain service from a central processing unit and so on. Moreover, retrial queues are also used as mathematical models for several computer systems: packet switching networks, shared bus local area networks operating under the carrier-sense multiple access protocol and collision avoidance star local area networks, etc. For more recent references see the bibliographical overviews in Artalejo [5-6]. Further, a comprehensive comparison between retrial queues and their standard counterpart with classical waiting line can be found in Artalejo and Falin [7]. The research in the area of retrial queues has focused mainly on the continuous-time setting. However, discrete-time retrial queues are more suitable to analyze some practical situations, such as WDM optical access networks, IP access networks and digital communication systems, because these systems operate on a discrete-time basis where events can only happen at regularly spaced epochs. Moreover, discrete-time models can be used to derive the results for continuous-time models but not vice versa (see Takagi [27]). Despite of the importance of the discrete-time retrial queues, little work has appeared concerning them. Using a generating function method, Li and Yang [22] considered the discrete-time Geo G 1 retrial queue for the first time, in which retrial times were assumed to have a geometric distribution and returning customers were assumed to act independently. For related literature on discrete-time retrial queues with geometric retrial times, the reader may refer to Artalejo et al. [8], Atencia and Moreno [9-10], Moreno [25], Wang and Zhao [28], Aboul-Hassan et al. [1], Liu and Gao [24] and references therein. Atencia and
Moreno [11] presented a discrete-time Geo G 1 retrial queue with general retrial times where only one customer may attempt retrials from orbit. Aboul-Hassan et al. [2-3] generalized Atencia and Moreno [11] to include balking customers and the batch arrival case, respectively. Wang and Zhao [29] generalized Atencia and Moreno [11] to the discrete-time Geo/G/1 retrial queue with general retrial times and starting failures. Gao et al. [15] considered a repairable discrete-time retrial queue with recurrent customers, Bernoulli feedback and general retrial times.
Recently, queueing systems with vacations have been studied extensively, comprehensive and excellent study on the vacation models, including some applications such as production/inventory system, communication systems, and computer systems, can be found in Gao and Liu [16], Jain et al. Li et al. [21] and other more literatures listed by Ke et al. [19]. For related literatures of retrial queues with vacations, the readers may refer to Artalejo [4], Krishna Kumar and Arivudainambi [20], Ke and Chang [18], Choudhury and Ke [13], and references therein for details. So far little work has appeared in the retrial queue with working vacations see Do [14], Li et al. [23] and Ayyappan et al. [12]. Do [14] discuss an M/M/1 retrial queue with working vacations and Li et al. [23] gives a Geo Geo 1 retrial queue with working vacations and vacation interruption. Using the matrix-analytic method, Do [14] and Li et al. [23] obtained the stationary probability distribution and some performance measures and showed the conditional stochastic decomposition for the queue length in the orbit. Similarly, Ayyappan et al. [12] considered a single server retrial queueing system with preemptive priority service and single working vacation by using Matrix geometric technique. However, in the literature there is no published work on Geo X G 1 queues with both general retrials and working vacations. In this paper, we introduce the new Geo X G 1 retrial queue with general retrial times, working vacations and vacation interruption. The rest of this paper is organized as follows. In Section 2, we give a brief description of the mathematical model. Section 3 provides the stable condition of the embedded Markov chain at the departure epochs. Section 4 deals with the steady state joint distribution of the server state and the number of customers in the orbit. And some important performance measures of this model are discussed briefly. Existence of the conditional stochastic decomposition for the queue length in the orbit is also demonstrated in Section 5. Some numerical results are presented in Section 6.
2. Mathematical Model Thereinafter, we denote x 1 x for any real number x (01) . The Geo X G 1 queue with general retrial times, working vacations and vacation interruption we considered here is an early arrival system, that is, a potential arrival and retrial can only take place in (m m ) and a potential departure can only take place in (m m ) . We assume that the beginning and ending of vacations occurs at the instant m . For this model, customers in the orbit are assumed to form a FCFS queue. It is supported by the following equivalent point of view: whenever the server becomes idle, it starts a process of search to find the next customer to be served. Various stochastic processes involved in the system are independent of each other. The various time epochs at which events occur are depicted in Figure 1. The detailed description of the model is given as follows: (1) Batches of customers arrive at the system according to a Bernoulli arrival process with parameter p , p is the probability that a batch of customers arrives in the interval (m m ) . The batch size sequence {X i }if 1 consists of independent and identically
*HR;*
distributed (iid) random variables distributed as X having the probability distribution function (pdf) {x i }if 1 , probability generating function (pgf) X ( z ) ¦ if 1 x i z i and n -th factorial moments [ n n 1 2 (2) We assume that there is no waiting space and therefore if the server is busy upon arrival, all the arriving customers join a pool of blocked customers called an orbit. If the server is not occupied, arriving customers get service immediately. We will assume that only the customer at the head of the orbit is allowed to access to the server and the inter-retrial times have an arbitrary distribution with pdf {ai }if 0 , pgf A( z ) ¦ if 0 ai z i . (3) The single server takes a working vacation at the epoch when customers being served depart from the system and no requests are in the system. The distribution of vacation time V is geometrically distributed with rate T (0 T d 1) , i.e., P (V j ) T T j 1, j t 1 . If there are customers arriving in a vacation period, the server continues to work at a lower rate. The working vacation period is an operation period at a lower speed. If there are customers in the system at a service completion instant in the vacation period, the server will come back to the normal working level, i.e., vacation interruption happens. Otherwise, the server continues the vacation. Meanwhile, if there is no customer when a vacation ends, the server begins another vacation, otherwise, the server switches to the normal working level and a regular busy period starts. At the end of a vacation if there is a customer whose service is not yet completed, then the server restart to serve the interruptive customer with normal service rate. (4) When the server is not on vacation, that is, the server is in busy period, the normal service time Sb has a general pdf P (Sb i ) sib i t 1 and pgf S b ( z ) ¦ if 1 sib z i and n -th factorial moments E n n t 1 (Obviously E1 E [Sb ] 1 / Pb ). While the service time Sv during a working vacation period has a general pdf P (Sv i ) siv i t 1 and pgf f v i S v ( z ) ¦ i 1 si z and mean E [Sv ] 1 Pv 0 Pv d Pb . At time m , the system can be described by the process Ym {J (m ) ] (m ) N (m )} , where N (m ) denotes the orbit size and J (m ) the state of the server,
J (m )
0, the server is on a working vacation at time m and the server is not occupied ° °1, the server is on a working vacation at time m and the server is busy ® °2, the server is not on a working vacation at time m and the server is busy °¯3, the server is not on a working vacation at time m and the server is not occupied.
] (m ) represents the remaining service time of the customer currently being served when J (m ) 1 2 and ] (m ) represents the remaining retrial time when J (m ) 3, N (m ) t 1 It can be shown that {Ym m N } is the Markov chain of our queueing system, whose state space is S {(0 0)} {(i j k ) i 1 2 j t 1 k t 0} {(3 j k ) j t 1 k t 1} In the following section, we will study the stability condition of the system by using embedded Markov chain method.
Figure 1. Various time epochs in an early arrival system.
3. Embedded Markov Chain and Stability Condition In this section, our objective is to give the stability condition of the system by using embedded Markov chain method. To construct a Markov process at embedded epochs, we consider the system at the slot boundaries immediately after a service completion. We denote the embedded epochs t1 t 2 " , i.e., t n be the slot of the n -th departure. For convenience, we assume that t 0 0 and let N n be the number of customers in the orbit at the slot boundary t n . Then the sequence of random variables {N n n t 0} forms a Markov chain with {01 2...} as state space and we have the following theorem. Theorem 1 The Markov chain {N n n t 0} is ergodic if and only if
U 1 [1 p (1 A( p))
(1)
Proof. To develop the transition matrix of {N n n t 0} . We firstly introduce the following denotations:
§m· m j smb ¨ ¸ p j p , k t 0, j 0 m max(1 j ) © j¹ k v § m · j m j m 1 ( j) ¦ xk ¦ sm ¨ ¸ p p T , k t 0, j 0 m max(1 j ) © j¹ f
k
Dk
( j) ¦ xk
bk
¦
k
( j) ¦ xk
vk
j 0
§m·
¦
m max(1 j )
T T m 1 ¨ ¸ p k p j ©
f
m j
¹
v ¦ si , k t 0,
i m 1
where x k( j ) is the probability that k customers arrive in j batches and is the j -fold convolution of x k , and x 0(0) 1 p pX ( z ) , the PGFs of {D k }fk 0 {bk }fk
Let K ( z )
D (z ) B(z ) V (z )
f
¦ Dk z
k
k 0 f
¦ bk z
k
f
m 1
m 1 f
¦ vk z
k
k 0
¦T
m 1
are given as follows:
S v (TK ( z )) ,
f
v m m 1 ¦ sm T ( p pX ( z ))
f
0
S b (K ( z )),
b m ¦ sm ( p pX ( z ))
k 0
and {v k }fk
0
T
f
m 1
T ( p pX ( z )) m ¦ siv i m 1
T (K ( z ) B ( z )) 1 TK ( z )
Evidently,
D (1)
f
¦ Dk
k 0
1 B (1)
f
¦ bk
k 0
S v (T ) V (1)
T
f
¦ vk
k 0
(T ) 1 Sv
T
Thus {E k k t 0} and {J k k t 0} are two non-complete probability distributions. Let c k
¦ kj 0 v j D k j k t 0 then
C (z )
f
¦ ck z
k 0
and
k
V ( z )D ( z ),
*HR;*
p[1
D c(1)
U B c(1)
Pb
B c(1) V c(1)
k
1 S v (T )
T
1 S v (T )
B c(1) C c(1)
p[1Svc (T ),
T
p[1
p[1 V (1) U
Furthermore, we define two sequence of probability distribution functions ^aˆk 0 1 2,...` and {bˆk k 01 2,...} , their generation functions are, respectively, given by f
Dl ( z ) l (z ) B
¦ Dˆ k z
f
b m 1 ¦ sm ( p pX ( z ))
k
k 0 f
k ¦ bˆk z
k 0
m 1 f
v m 1 m 1 ¦ sm T ( p pX ( z ))
m 1
D (z ) , K(z ) B(z ) K(z)
Put cˆk ¦ kj 0 v j Dˆ k j k t 0 and define Pij P ( N n 1 j _ N n i ) , the Pij ’s are the one step transition probabilities from i to j . Then the transition probability matrix Q of the embedded Markov chain {N n n t 0} can be written as
Q
ª P00 « « P10 « « « « «¬
P01
P02
P11
P12
P21
P22 P32
P03 "º » P13 "» P23 "» . » P33 "» » # # »¼
(2)
It is not difficult to see that {N n n t 0} is irreducible and aperiodic. Noting that working vacation and vacation interruption policy is introduced into our queueing system, then if there are customers in the orbit immediately after one customer’s departure epoch (no matter whether served by normal service rate or by lower service rate), the next customer (external or repeated) will be served by the normal service rate, so we have Pi j
j i
pA( p )Dˆ j i 1 (1 G j i 1 )(1 pA( p )) ¦ x m 1Dˆ j i m i t 1 j t i 1,
(3)
m 0
where G j i 1 denotes the Kronecker delta. We assume that the beginning and ending of vacations occurs at the instant m , if there is no customer in the orbit at the slot boundary (say tn ) immediately after one departure, we should consider the following two cases: (1) there is a batch of customers arrives in the interval (t n t n ) , in this case, the server will provide the normal service for one of the batch customers (selected at random) and the others enter into the orbit; (2) no customer arrives in the interval (t n t n ) , and then the server begins a working vacation. In this case, we should distinguish between (T V ) and (T V ) , where T denotes the inter-arrival time. When (T V ) , the server will serve the next customer at
!!
lower service rate because the vacation doesn’t end at the arrival epoch; when (T server will serve the next customer at normal service rate. We can calculate that
P (T V )
pT 1 pT
P (T
pT 1 pT
V)
V ) , the
.
Because customers arrive in bulk, then using the probabilistic argument, we have P0 k
k § P (T V ) k · P (T V ) k p ¦ x m 1Dˆ k m p ¨ ¦ x m 1Dˆ k m ¸ ¦ x m 1 (b k m cˆk m ) P (T d V ) m 0 m 0 © P (T d V ) m 0 ¹ k k k p ¦ x m 1Dˆ k m p §¨ T ¦ x m 1 (b k m cˆk m ) T ¦ x m 1Dˆ k m ·¸ , k t 0 m 0 m 0 ¹ © m 0
(4)
To investigate ergodicity of the Markov chain {N n n t 0} , we use Foster’s criterion based on the mean drift xi
E [ f ( N n 1 ) f ( N n ) _ N n
i ]
where the test function f (i ) i i t 0 . For i t 1 it can be easily proved that, xi
E [ f ( N n 1 ) f ( N n ) _ N n
i]
U 1 [1 p (1 A( p )) While for i t 0 , x0
E [ f ( N n 1 ) f ( N n ) _ N n
U 1 [1 p (1
0]
pT (1 S v (T ))
T
p E1 S v (T ))
From the transition matrix, we know that the mean drift is given by x i U 1 [1 p (1 pT (1 S v (T )) T p E1 S v (T )) if i=0; U 1 [1 p (1 A( p )) if i t 1 . Clearly, the inequality U 1 [1 p (1 A( p)) is a sufficient condition for ergodicity.
The same inequality is also necessary for ergodicity. As noted in Sennot et al. [26], we can guarantee nonergodicity if the Markov chain {N n n t 0} satisfies Kaplan’s condition, namely, x i f for all i t 0 and there exists i0 N such that x i t 0 for all i t i0 . Notice that, in our case, Kaplan’s condition is fulfilled because Pij 0 for j d i 1 and i ! 0 . Then, U t 1 [1 p (1 A( p )) implies the nonergodicity of the Markov chain. Let’s observe that the condition U 1 [1 p (1 A( p)) can be rewritten as p[1 ( E1 1) (1 pA( p ))([1 1) d pA( p ) , where the first member is the expected number of external customers who arrive into the orbit per normal service interval and the second one denotes the expected number of a batch of external customers who enter into the orbit when the batch customers arrive before the orbiting customers, the third one represents the expected number of repeated customers who enter service at the epoch at which a normal service starts. Therefore, this condition indicates that external customers (the first member plus the second one) must arrive more slowly than repeated customers can enter normal service at the epoch at which a normal service starts (on the average). Accordingly, if the condition U 1 [1 p (1 A( p)) is fulfilled, then the system is stable and consequently this condition is also sufficient. On the other hand, this condition is a necessary condition for
*HR;* !
the ergodicity of the Markov chain Ym {J (m ) ] (m ) N (m ) m N } since P00 ! 0 (the exact expression of P00 is obtained in Section 4). Throughout this paper, we assume the stationary condition U 1 [1 p (1 A( p )) is satisfied.
4. Steady State Distribution of the System In this section, we first develop the steady-state equations for the retrial system by treating the remaining retrial time, the remaining time of the normal/lower service as supplementary variables. Then we derive the probability generating functions for the server state and the number of customers in the system/orbit.
Define lim P ( J (m ) 0 N (m ) 0)
P00 Pj i k
lim P ( J (m )
m of
m of
j ] (m ) i N (m )
k ) j 1 2 i t 1 k t 0 j
3 i t 1 k t 1
Following the routine procedure of the method of supplementary variables, we have the following set of equilibrium equations P00
pP00 pP110 pP210
P1i k
T ¨¨ px k 1 siv P00 pP1i 1k (1 G k 0 ) p ¦ x m P1i 1k m ¸¸ i t 1 k t 0
P2i k
T sib ¨¨ p ¦ P1 j k (1 G k 0 ) p ¦ ¦ x m P1 j k m ¸¸ pT x k 1 sib P00 ¨ ¸
(5)
§
k
·
©
m 1
¹
§
f
k 1 f
·
©
j 2
m 1j 2
¹
§
k
·
k
©
m 0
¹
m 1
§
k
·
©
m 0
¹
(6)
¨¨ pa0 P11 k 1 p ¦ x m 1 P11k m ¸¸ sib pP2i 1 k (1 G k 0 ) p ¦ x m P2i 1k m ¨¨ pa0 P21k 1 p ¦ x m 1 P21 k m ¸¸ sib psib P31 k 1 k 1
f
m 0
j 1
(1 G k 0 ) psib ¦ x m 1 ¦ P3 j k m i t 1 k t 0
(7)
p P3i k 1 ai P11 k P21 k i t 1 k t 1
(8)
P3i k
and the normalization condition is f
f
f f
P00 ¦ ¦ P1i k P2i k ¦ ¦ P3i k i 1k 0
i 1k 1
1
In order to obtain the solution of (5)-(8), we define the following generating functions: ) j (x z )
f
f
i k ¦ ¦ Pj i k x z j
1 2 b
0 j
3 b 1
1 2 b
0 j
3 b 1
i 1k b
and the auxiliary generating functions ) j i ( z )
f
k ¦ Pj i k z i t 1 j
k b
!"
Multiplying (6-8) by z k and summing over k leads to )1i ( z ) TK ( z ))1i 1 ( z )
T pX ( z ) z
siv P00 i t 1
sib (TK ( z )()1 (1 z ) )11 ( z ))
) 2i ( z )
(9)
pX ( z ) pa0 ()11 ( z ) ) 21 ( z )) z
pa0 pT X ( z ) pX ( z ) p P00 ( P110 P210 ) ) 3 (1 z ) ) 31 ( z )) z z z z
(10)
K ( z )) 2i 1 ( z ) i t 1 ) 3i ( z )
p () 3i 1 ( z ) ai ()11 ( z ) ) 21 ( z ) ( P110 P210 ))) i t 1
(11)
Multiplying (9) by x i and summing over i yields x TK ( z ) )1 ( x z ) x Choosing x we have
pT X ( z ) S v ( x ) P00 TK ( z ))11 ( z ) z
(12)
TK ( z ) in (12), the left hand side of this equation vanishes and therefore )11 ( z )
pX ( z ) E ( z ) P00 zK ( z )
(13)
where E ( z ) S v (TK ( z )) Substituting the above expression into (12), we can get )1 ( x z )
S v ( x ) E ( z ) X ( z ) T pxP 0 0 z x TK ( z )
(14)
Similarly, multiplying (10)-(11) by z k and x i , summing over k and i and using (5) leads to pX ( z ) pa0 x K (z ) ()11 ( z ) ) 21 ( z )) TK ( z ))11 ( z ) ) 2 ( x z ) S b ( x )(TK ( z ))1 (1 z ) x z
T pX ( z ) pa0 z
P00
pX ( z ) p ) 3 (1 z ) ) 31 ( z )) K ( z )) 21 ( z ) z z
xp ) 3 ( x z ) ( A( x ) a0 )( p ()11 ( z ) ) 21 ( z )) pP00 ) p ) 31 ( z ) x
(15) (16)
Putting x 1 in (14) yields (1 TK ( z )))1 (1 z )
T pX ( z )(1 E ( z )) z
P00
(17)
Before finding ) 21 ( z ) ) 31 ( z ) ) 3 (1 z ) , we first introduce one lemma, whose proofs can be readily obtained.
Lemma 1 (1) D ( z )( pA( p ) (1 pA( p )) X ( z )) zK ( z ) ! 0 holds for 0 d z 1 if and only if U 1 [1 p
u(1 A( p )) .
*HR;* !# (2) The following limits exist if and only if U 1 [1 p (1 A( p)) :
lim z o1
zK ( z )(T X ( z )(1 V ( z )) A( p )(1 X ( z ))) X ( z ) E ( z )( pA( p ) (1 pA( p )) X ( z )) D ( z )( pA( p ) (1 pA( p )) X ( z )) zK ( z )
T[1 A( p ) ( pT[1 (1 A( p )) (T p[1T ))S v (T ) p[1T , T (1 U [1 p (1 A( p ))) lim
zK ( z ) X ( z )( p E ( z ) T (C ( z ) D ( z )) D ( z ))
D ( z )( pA( p ) (1 pA( p )) X ( z )) zK ( z )
z o1
T ( p[1 U 1) p ( p[1T (1 S v (T )) TU S v (T ))) T (1 U [1 p (1 A( p ))) To find ) 21 ( z ) ) 31 ( z ) and ) 3 (1 z ) , we first set x
p in (16), then
p ) 31 ( z ) ( A( p ) a0 )( p ()11 ( z ) ) 21 ( z )) pP00 ). Substituting (18) into (16) and putting x p) 3 (1 z )
(18)
1 yields
(1 A p )( p ()11 ( z ) ) 21 ( z )) pP00 ).
(19)
Inserting )11 ( z ) )1 (1 z ) ) 31 ( z ) ) 3 (1 z ) in (15), then x K (z ) )2 ( x z ) x
pS x I z ( pA( p ) (1 pA( p )) X ( z ))Sb ( x ) zK ( z ) ) 21 ( z ) b P0,0 , (20) z z
where I ( z ) T X ( z )(1 V ( z )) A( p )(1 X ( z )) X ( z ) E ( z )( pA( p ) (1 pA( p )) X ( z ))( zK ( z )) 1 Taking x K ( z ) in (20), and then after some calculations we can obtain that ) 21 ( z )
pD ( z ) I ( z ) P D ( z )( pA( p ) (1 pA( p )) X ( z )) zK ( z ) 00
(21)
Inserting (13) and (21) into (16) and (20), we can obtain )2 ( x z )
S b ( x ) D ( z ) px P u 0 0 x K (z ) z
zK ( z )(T X ( z )(1 V ( z )) A( p )(1 X ( z ))) X ( z ) E ( z )( pA( p ) (1 pA( p )) X ( z )) (22) D ( z )( pA( p ) (1 pA( p )) X ( z )) zK ( z ) and )3 ( x z )
A( x ) A( p ) zK ( z ) X ( z )( p E ( z ) T (C ( z ) D ( z )) D ( z )) pxP00 . xp D ( z )( pA( p ) (1 pA( p )) X ( z )) zK ( z )
(23)
Using the normalization condition P00 )1 (11) )1 (11) ) 3 (11) 1 , we get the probability that the system is empty as follows P00
T (1 U p[1 (1 A( p ))) T A( p ) p (T (1 S v (T )) TE1 S v (T ))
(24)
!
Obviously from (24), as P00 ! 0 , we obtain that U 1 p[1 (1 A( p ))) is a necessary condition for the ergodicity of the Markov chain Ym {J (m ) ] (m ) N (m ) m N } . Now we summarize the above results in the following theorem.
Theorem 2 The generating functions of the stationary joint distribution of the Markov chain {Ym m N } are given by: S v ( x ) E ( z ) X ( z ) T pxP , 00 z x TK ( z ) ( x ) D ( z ) px )2 ( x z ) S b P00 u x K (z ) z
)1 ( x z )
zK ( z )(T X ( z )(1 V ( z )) A( p )(1 X ( z ))) X ( z ) E ( z )( pA( p ) (1 pA( p )) X ( z )) D ( z )( pA( p ) (1 pA( p )) X ( z )) zK ( z ) )3 ( x z )
A( x ) A( p ) zK ( z ) X ( z )( p E ( z ) T (C ( z ) D ( z )) D ( z )) pxP00 D ( z )( pA( p ) (1 pA( p )) X ( z )) zK ( z ) xp
where P00 is given by (24). Next we are interested in the generating functions of some important distributions.
Corollary 1 (1) The marginal generating function of the orbit size when the server is busy with lower rate service is: 1 E (z ) X (z ) )1 (1 z ) T pP00 1 TK ( z ) z (2) The marginal generating function of the orbit size when the server is busy with normal
service is: ) 2 (1 z )
zK ( z )(T X ( z )(1 V ( z )) A( p )(1 X ( z ))) X ( z ) E ( z )( pA( p ) (1 pA( p )) X ( z )) D ( z )( pA( p ) (1 pA( p )) X ( z )) zK ( z ) u
1 D (z ) p0 0 z (1 X ( z ))
(3) The marginal generating function of the orbit size when the server is not on a working
vacation and is available:
) 3 (1 z )
zK ( z ) X ( z )( p E ( z ) T (C ( z ) D ( z )) D ( z ))
D ( z )( pA( p ) (1 pA( p )) X ( z )) zK ( z )
(4) The pgf of the orbit size, denoted as < ( z ) , is given by
