Jrl Syst Sci & Complexity (2008) 21: 304–315
ON THE SINGLE SERVER RETRIAL QUEUE WITH PRIORITY SUBSCRIBERS AND SERVER BREAKDOWNS∗ Jinting WANG
Received: 10 April 2007 / Revised: 21 December 2007 c °2008 Springer Science + Business Media, LLC Abstract The author concerned the reliability evaluation as well as queueing analysis of M1 , M2 /G1 , G2 /1 retrial queues with two different types of primary customers arriving according to independent Poisson flows. In the case of blocking, the first type of customers can be queued whereas the second type of customers must leave the service area but return after some random period of time to try their luck again. The author assumes that the server is unreliable and it has a service-type dependent, exponentially distributed life time as well as a service-type dependent, generally distributed repair time. The necessary and sufficient condition for the system to be stable is investigated. Using a supplementary variable method, the author obtains a steady-state solution for queueing measures, and the transient as well as the steady-state solutions for reliability measures of interest. Key words Priority queues, reliability, retrial queues, server breakdowns.
1 Introduction Retrial queueing systems are characterized by the feature that arrivals who find the server unavailable are obliged to leave the service area and to try again for their requests in random order and at random intervals. Between trials a customer is called to be in “orbit”. This feature plays a special role in several computer and communications networks. For recent bibliographies on retrial queues, see [1–4]. Artalejo[5−6] also provided extensive surveys of retrial queues. The single server retrial queues with priority calls have been studied by a number of researchers. Such queueing systems arise naturally as practical models in daily life and in communication networks, for example, making reservations, packet switching networks, non-persistent CSMA (Carrier Sense Multiple Access) and real time systems. In a recent paper [7], Choi and Chang gave these examples of practical situations where such models arose and presented the detailed survey of the results concerning these models. Choi and Park[8] investigated a retrial queue with two types of calls, infinite priority queue for type-1 calls and infinite retrial group of type-2 calls. They obtained the joint generating function of queue lengths by supplementary variable method. This model turned out to be essentially identical to the retrial queue with two-way communication and infinite sources of Jinting WANG Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China. Email:
[email protected]. ∗ This research is sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, Education Ministry and the National Natural Science Foundation of China under Grant Nos 10526004 and 60504016.
ON THE SINGLE SERVER RETRIAL QUEUE
305
outgoing calls investigated by Falin[2] . Later on, Falin et al.[9] extended Choi and Park’s results to the case that two types of calls may have different service time distributions. It is noted that most papers on retrial queues assume that the server is available on a permanent basis. In practice, however, these assumptions are apparently unrealistic. The server may well be subject to lengthy and unpredictable breakdowns while serving a customer. For example, in computer systems, the machine may be subject to scheduled backups and unpredictable failures. Because of limited ability of repairs and heavy influence of the breakdowns on the performance measure of the system, it is of essential importance to study reliability of retrial queues with server breakdowns and repairs. Wang et al.[10] carried out a detailed analysis of reliability of the classic M/G/1 retrial queues with exponentially distributed retrial times. For related literature, interested readers may refer to [11–14], and references therein. In this paper, we discuss the M1 , M2 /G1 , G2 /1 retrial queues with priority subscribers and server subject to breakdowns and repairs. Such unreliable systems also arise from practical situations in daily life and in communication networks. For instance, consider subscriber line modules of telephone exchanges with undetected breakdowns[15] , or fax machine operation with unreliable fax machines[7] . Our study generalizes the models studied by Falin et al.[9] and Wang et al.[10] .
2 Model Description Consider a single-server retrial queue in which two different types of primary customers arrive according to independent Poisson process flows with rates λ1 and λ2 , respectively. Service times for customers from the type-i flow are independent and identically distributed positive random variables with a common distribution function Bi (x), density function bi (x), Laplaceei (s), and the first two moments βi1 and βi2 (i = 1, 2). If the server is free Stieltjes transform B at the time of any primary call arrival, this call begins to be served immediately and leaves the system after service completes. We assume that the server is subject to random breakdowns when it works, and it has a service-type dependent, exponentially distributed life time as well as a service-type dependent, generally distributed repair time. That is, the server’s life time has exponential distribution with mean α11 when it serves type-1 customers, and when type-2 customers receive their service, the server fails at an exponential rate α2 . Once the busy server fails, repair begins immediately. If the server fails when serving type-i customers, the time required to repair is a random variable with probability distribution function Gi (x), density function gi (x), Laplace-Stieltjes transform e i (x), and with the first two moments γi1 and γi2 (i = 1, 2). Furthermore, we assume that the G service time for a customer is cumulative, and the server works as a new one after repair. So the state of the server may be free, busy, or under repair. If the server is found to be busy or under repair, behavior of the arriving primary customer depends on its type. Customers from the first flow are queued after blocking and then are served in some discipline such as FCFS or random order. Customers from the second flow who find the server unavailable upon arrival cannot be queued and leave the service area, but after some random delay repeat an attempt to get service. We assume that intervals between retrials are exponentially distributed with parameter µ. It is easy to see that a type-2 customer can be admitted for service only if there is no queue of type-1 customers. Thus the type-1 customers have a non-preemptive priority over the type-2 customers. When the retrial rate tends to infinity, the model under consideration can be thought of as the standard queueing system with head of line priority discipline and server breakdowns. For convenience, we refer to the type-1 subscribers as priority customers and the type-2 subscribers as non-priority customers.
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As usual, we assume that the inter-arrival times of primary calls, retrial times, service times, and repair times are mutually independent. At time t, let N1 (t) and N2 (t) be the number of customers in priority and non-priority queue, respectively; C(t) represents the server state and equals to 0, 1, 2, 3, or 4 according to the server which is free, is occupied by a type-1 customer, is occupied by a type-2 customer, fails when serving a type-1 customer, or fails when serving a type-2 customer, respectively. The process is not Markovian because the service time and repair time distribution are not exponential. To make it Markovian, we introduce two supplementary variables: if C(t) = 1, 2, 3, 4, we define ξ1 as the elapsed service time of the call being served; if C(t) = 3, 4, we define ξ2 as the elapsed repair time of the server being repaired. Let ρi be the system load due to ith type of primary calls, that is, ρi = λi βi,1 and ρ = ρ1 +ρ2 . Bi0 (x) be the instantaneous service intensity when type-i customers receive service Let βi (x) = 1−B i (x) G0 (x)
i given that the elapsed service time is equal to x, and γi (x) = 1−G be the instantaneous repair i (x) intensity when type-i breakdowns happen given that the elapsed repair time is equal to y. From the description of the model, the state of the system at time t can be described by the Markov process {X(t), t ≥ 0} = {(C(t), N1 (t), N2 (t), ξ1 (t), ξ2 (t)); t ≥ 0}.
3 Stability Condition and the Steady-State Solutions To prove ergodicity, we need the following preliminary results. It should be noted that the “true” service time is not the length of time from the epoch when a customer begins to be served until the service is completed because of possible breakdowns. To make it clear, we define the generalized service time as the length of time from when a customer begins to be served until the service is completed. Let Tin be the generalized service time of customer i in the nth type. We note that it may include some possible down times of the server due to server failures during the service period of the nth type-i customer, since the nth type-i customer begins to be served until the service is completed. It was shown in [16] that Tin s are i.i.d. random variables with distribution function ∆
Di (t) = Pr{Tin ≤ t} =
∞ Z X l=0
0
t
(l)
Gi (t − u)e−αi u
(αi u)l dBi (u), l!
i = 1, 2,
which is independent of n. Its Laplace-Stieltjes transform is Z ∞ e ei (s + αi − αi G e i (s)), Di (s) = e−st dDi (t) = B 0
and its expected value is given by ETin = −
e i (s) ¯¯ dD = βi1 (1 + αi γi1 ). ¯ ds s=0
(3.1)
Let ηd be the time of the dth departure. Then the sequence of random vectors Xd = (C(ηd − 0), N1 (ηd − 0), N2 (ηd − 0)) forms an embedded Markov chain with state space Ω = 2 {1, 2, 3, 4} × Z+ . By the classical Foster’s criteria with the Lyapunov’s function f (s), s ∈ Ω , such as ∆
f (k, n, m) = [λ2 β1,1 (1 + α1 γ1,1 ) + 1 − λ2 β2,1 (1 + α2 γ2,1 )]m +[λ1 β2,1 (1 + α2 γ2,1 ) + 1 − λ1 β2,1 (1 + α2 γ2,1 )]n,
307
ON THE SINGLE SERVER RETRIAL QUEUE
the mean drift of the embedded chain from state (k, n, m) is given by xk,m,n = E[f (Xd ) − f (Xd−1 ) | Xd−1 = (k, m, n)] ρ1 (1 + α1 γ1,1 ) + ρ2 (1 + α2 γ2,1 ) − 1, = ρ1 (1 + α1 γ1,1 ) + ρ2 (1 + α2 γ2,1 ) − 1 +
if m ≥ 1, λ1 + λ 2 , λ1 + λ2 + nµ
if m = 0.
Thus, applying the classic Foster criteria we can provide the following theorem to give a necessary and sufficient condition for the system to be stable. Theorem 3.1 The inequality ρ1 (1+α1 γ1,1 )+ρ2 (1+α2 γ2,1 ) < 1 is a necessary and sufficient condition for the system to be stable. The proof can be adapted from [9]. It just needs replacing service time by the generalized service time and therefore we omit it here. e i (λ1 − λ1 z1 + λ2 − λ2 z2 ). It is easy to see that For convenience, we let ki (z1 , z2 ) = D ki (z1 , z2 ) =
∞ X ∞ X
ki,m,n z1m z2n ,
m=0 n=0
where
Z ki,m,n = 0
∞
(λ1 x)m −λ1 x (λ2 x)n −λ2 x e e dDi (x) m! n!
is the joint distribution of the number of primary calls of both types which arrive during the generalized service time of the ith-type call. In the sequel, we shall study the system in steady state. Hence, the condition ρ1 (1+α1 γ1,1 )+ ρ2 (1 + α2 γ2,1 ) < 1 is assumed to hold from now. We introduce the following functions which describe the joint distribution of the server state and queue length in the steady state: p0ij (t) ≡ P {C(t) = 0, N1 (t) = i, N2 (t) = j}, d p1,i,j (t, x) = P {C(t) = 1, ξ1 < x, N1 (t) = i, N2 (t) = j}, dx d p2,i,j (t, x) = P {C(t) = 2, ξ1 < x, N1 (t) = i, N2 (t) = j}, dx d p3,i,j (t, x, y) = P {C(t) = 3, ξ1 < x, ξ2 (t) < y, N1 (t) = i, N2 (t) = j}, dy d p4,i,j (t, x, y) = P {C(t) = 4, ξ1 < x, ξ2 (t) < y, N1 (t) = i, N2 (t) = j}, dy and under the stability condition we denote the steady state probabilities as: p0ij = lim p0ij (t), t→∞
p1ij (x) = lim p1ij (t, x), t→∞
p3ij (x, y) = lim p3ij (t, x, y), t→∞
p1ij (x) = lim p1ij (t, x), t→∞
p4ij (x, y) = lim p4ij (t, x, y). t→∞
In a general way, we obtain the following equations of statistical equilibrium: Z ∞ (λ1 + λ2 + jµ)p00j = [p10j (x)β1 (x) + p20j (x)β2 (x)]dx, j ≥ 0,
(3.2)
0
p0ij = 0,
if i ≥ 1, j ≥ 0
(3.3)
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p01ij (x) = −[λ1 + λ2 + β1 (x) + α1 ]p1ij (x) + λ1 p1,i−1,j (x) Z ∞ + λ2 p1,i,j−1 (x) + p3ij (x, y)γ1 (y)dy, i, j ≥ 0,
(3.4)
0
p02ij (x) = −[λ1 + λ2 + β2 (x) + α2 ]p2ij (x) + λ1 p2,i−1,j (x) Z ∞ + λ2 p2,i,j−1 (x) + p4ij (x, y)γ2 (y)dy, i, j ≥ 0,
(3.5)
0
∂p3ij (x, y) = −[λ1 + λ2 + γ1 (y)]p3ij (x, y) + λ1 p3,i−1,j (x, y) ∂y + λ2 p3,i,j−1 (x, y), i, j ≥ 0, ∂p4ij (x, y) = −[λ1 + λ2 + γ2 (y)]p4ij (x, y) + λ1 p4,i−1,j (x, y) ∂y + λ2 p4,i,j−1 (x, y), i, j ≥ 0. Boundary conditions are as follows: Z p1ij (0) = δ0i λ1 p00j +
Z
∞
(3.7)
∞
p1,i+1,j (x)β1 (x)dx +
0
(3.6)
p2,i+1,j (x)β2 (x)dx,
(3.8)
0
p2ij (0) = δ0i [λ2 p00j + (j + 1)µp0,0,j+1 ], p3ij (x, 0) = α1 p1ij (x), p4ij (x, 0) = α2 p2ij (x),
(3.9) (3.10) (3.11)
where δ0j is the Kronecker function, and for any fixed t, x, and y, ∆
p1,−1,j (x) = p1,i,−1 (x) = p2,−1,j (x) = p2,i,−1 (x) = 0, ∆
p3,−1,j (x, y) = p3,i,−1 (x, y) = p4,−1,j (x, y) = p4,i,−1 (x, y) = 0. These equations are to be solved by using the generating function technique. To this end, we define the corresponding partial generating functions: p0 (z1 , z2 ) =
∞ ∞ X X
z1i z2j p0ij ,
i=0 j=0 ∞ ∞ X X
p2 (z1 , z2 , x) =
z1i z2j p2ij (x),
i=0 j=0 ∞ X ∞ X
p4 (z1 , z2 , x, y) =
p1 (z1 , z2 , x) =
∞ ∞ X X
z1i z2j p1ij (x),
i=0 j=0 ∞ ∞ X X
p3 (z1 , z2 , x, y) =
z1i z2j p3ij (x, y),
i=0 j=0
z1i z2j p4ij (x, y),
|z1 | ≤ 1, |z2 | ≤ 1
i=0 j=0
Similar to the derivation by Falin and Templeton[4] , we obtain the following results. Details are omitted due to the space limitation. Theorem 3.2 In the steady state, the joint distribution of the server state and queue length has partial generating functions: p0 (z2 ) ≡ p0 (z1 , z2 ) = [1 − ρ1 (1 + α1 γ1,1 ) − ρ2 (1 + α2 γ2,1 )] ¾ ½ Z z2 λ1 − λ1 h(u) + λ2 − λ2 k2 (h(u), u) 1 du , × exp µ 1 k2 (h(u), u)
(3.12)
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309
© p1 (z1 , z2 , x) = (λ1 − λ1 h(z2 ) + λ2 − λ2 k2 (h(z2 ), z2 ))(k2 (z1 , z2 ) − z2 ) ª − (λ1 − λ1 z1 + λ2 − λ2 k2 (z1 , z2 ))(k2 (h(z2 ), z2 ) − z2 ) ×[(k2 (h(z2 ), z2 ) − z2 )(z1 − k1 (z1 , z2 ))]−1 ×p0 (z2 )[1 − B1 (x)]e−Φ1 (z1 ,z2 )x , λ1 − λ1 h(z2 ) + λ2 − λ2 z2 × p0 (z2 )[1 − B2 (x)]e−Φ2 (z1 ,z2 )x , k2 (h(z2 ), z2 ) − z2 n p3 (z1 , z2 , x, y) = α1 (λ1 − λ1 h(z2 ) + λ2 − λ2 k2 (h(z2 ), z2 ))(k2 (z1 , z2 ) − z2 ) o − (λ1 − λ1 h(z2 ) + λ2 − λ2 k2 (z1 , z2 ))(k2 (z1 , z2 ) − z2 ) p2 (z1 , z2 , x) =
(3.13) (3.14)
×[(k2 (z1 , z2 ) − z2 )(z1 − k1 (z1 , z2 ))]−1 ×p0 (z2 )[1 − B1 (x)][1 − G1 (y)]e−Φ1 (z1 ,z2 )x−(λ1 −λ1 z1 +λ2 −λ2 z2 )y , p4 (z1 , z2 , x, y) =
(3.15)
α2 (λ1 − λ1 h(z2 ) + λ2 − λ2 z2 ) k2 (h(z2 ), z2 ) − z2 ×p0 (z2 )[1 − B2 (x)][1 − G2 (y)]e−Φ2 (z1 ,z2 )x−(λ1 −λ1 z1 +λ2 −λ2 z2 )y ,
(3.16)
where £ ¤ e 1 (λ1 − λ1 z1 + λ2 − λ2 z2 ) , Φ1 (z1 , z2 ) = λ1 − λ1 z1 + λ2 − λ2 z2 + α1 1 − G £ ¤ e 2 (λ1 − λ1 z1 + λ2 − λ2 z2 ) , Φ2 (z1 , z2 ) = λ1 − λ1 z1 + λ2 − λ2 z2 + α2 1 − G e 1 (z1 , z2 )]. and h(z2 ) is the unique root in the desk |z2 | ≤ 1 of the equation z1 = B[Φ Corollary 3.3 If ρ1 (1 + α1 γ1,1 ) + ρ2 (1 + α2 γ2,1 ) < 1, then 1) The probability that the system is empty: Pempty = 1 − ρ1 (1 + α1 γ1,1 ) − ρ2 (1 + α2 γ2,1 ); 2) The probability that the server is busy: Pbusy = ρ1 + ρ2 ; 3) The probability that the server is under repair: Prepair = ρ1 α1 γ1,1 + ρ2 α2 γ2,1 . Corollary 3.4 In the steady state, if we neglect the elapsed service time and repair time, the joint distribution of the number of priority queue and non-priority queue plij = P {C(t) = P∞ l, N1 (t) = i, N2 (t) = j} ≡ i=0 z1i z2j plij has the following generating functions (l = 1, 2, 3, 4): © p1 (z1 , z2 ) = (λ1 − λ1 h(z2 ) + λ2 − λ2 k2 (h(z2 ), z2 ))(k(z1 , z2 ) − z2 ) ª − (λ1 − λ1 z1 + λ2 − λ2 k2 (z1 , z2 ))(k2 (h(z2 ), z2 ) − z2 ) ×[(k2 (h(z2 ), z2 ) − z2 )(z1 − k1 (z1 , z2 ))]−1 ×
p2 (z1 , z2 ) =
1 − k1 (z1 , z2 ) p0 (z2 ), Φ1 (z1 , z2 )
λ1 − λ1 h(z2 ) + λ2 − λ2 z2 1 − k2 (z1 , z2 ) · p0 (z2 ), k2 (h(z2 ), z2 ) − z2 Φ2 (z1 , z2 )
(3.17)
(3.18)
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JINTING WANG
n p3 (z1 , z2 ) = α1 (λ1 − λ1 h(z2 ) + λ2 − λ2 k2 (h(z2 ), z2 ))(k2 (z1 , z2 ) − z2 ) o − (λ1 − λ1 h(z2 ) + λ2 − λ2 k2 (z1 , z2 ))(k2 (z1 , z2 ) − z2 ) ×[(k2 (z1 , z2 ) − z2 )(z1 − k1 (z1 , z2 ))]−1 × p4 (z1 , z2 ) =
e 2 (λ1 − λ1 z1 + λ2 − λ2 z2 ) 1 − k1 (z1 , z2 ) 1 − G × p0 (z2 ), Φ1 (z1 , z2 ) λ 1 − λ 1 z1 + λ 2 − λ 2 z2
α2 (λ1 − λ1 h(z2 ) + λ2 − λ2 z2 ) k2 (h(z2 ), z2 ) − z2 e 2 (λ1 − λ1 z1 + λ2 − λ2 z2 ) 1 − k2 (z1 , z2 ) 1 − G × × p0 (z2 ). Φ2 (z1 , z2 ) λ 1 − λ 1 z1 + λ 2 − λ 2 z2
(3.19)
(3.20)
In particular, the mean queue length in the priority and non-priority queue are given by EN1 (t) =
λ1 [λ1 α1 β1,1 γ1,2 + λ1 (1 + α1 γ1,1 )2 β1,2 + λ2 α2 β2,1 γ2,2 + λ2 (1 + α2 γ2,1 )2 β2,2 ] 2[1 − ρ1 (1 + α1 γ1,1 )]
and EN2 (t) =
λ2 [ρ1 (1 + α1 γ1,1 ) + ρ2 (1 + α2 γ2,1 )] µ[1 − ρ1 (1 + α1 γ1,1 ) − ρ2 (1 + α2 γ2,1 )] +
λ2 [λ1 α1 β1,1 γ1,2 + λ1 (1 + α1 γ1,1 )2 β1,2 + λ2 α2 β2,1 γ2,2 + λ2 (1 + α2 γ2,1 )2 β2,2 ] . 2(1 − ρ1 (1 + α1 γ1,1 ))[1 − ρ1 (1 + α1 γ1,1 ) − ρ2 (1 + α2 γ2,1 )]
Remark 3.5 When α1 = α2 = 0, our model becomes the M1 , M2 /G1 , and G2 /1 retrial queue with reliable server. These results are consistent with known results in [4,9]. When λ1 = 0, our model becomes the M/G/1 retrial queue with server breakdowns and repairs studied in [10]. It is easy to check that our results are consistence with corresponding results in the above-mentioned papers, and we omit the detailed deduction here.
4 Reliability Indexes of the Server We now consider some reliability quantities of the server. Let A(t) = P {the service station is up at time t}, which is defined as the point-wise availability of the server, and we define the steady-state availability of the server as A = lim A(t). t→∞ Theorem 4.1 The steady-state availability of the server is A = 1 − ρ1 α1 γ1,1 − ρ2 α2 γ2,1 . Proof This is readily obtained by considering the following equation: A=
∞ X ∞ X
p0ij +
i=0 j=0
∞ X ∞ Z X i=0 j=0
∞
[p1ij (x) + p2ij (x)]dx
0
½ Z = lim lim p0 (z1 , z2 ) + z1 →1 z2 →1
0
∞
¾ [p1 (z1 , z2 , x) + p2 (z1 , z2 , x)]dx ,
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311
together with (3.12)–(3.14). Theorem 4.2 The steady-state failure frequency of the server is Wf = α1 ρ1 + α2 ρ2 . Proof Since the steady-state failure frequency of the service station is ∞ X ∞ Z +∞ X Wf = [α1 p1ij (x) + α2 p2ij (x)]dx, i=0 j=0
0
we get Z Wf = lim lim
z1 →1 z2 →1
+∞
[α1 p1 (z1 , z2 , x) + α2 p2 (z1 , z2 , x)]dx = α1 ρ1 + α2 ρ2 . 0
Denote by τ the time to the first failure of the server. The reliability function of the server is defined as R(t) = P (τ > t) . Theorem 4.3 The Laplace-Stieltjes transform of R(t) is given by R∗ (s) = D1 (s) + where
Z
D1 (s) = 1
w(s)
e1 (s + λ1 ) e2 (s + λ2 ) 1−B 1−B D2 (s) + D3 (s), s + λ1 s + λ2
(4.1)
1 e µ[B2 (s + λ1 + λ2 + α2 − λ1 r(y) − λ2 y) − y]
( Z ) e2 (s+λ1 +λ2 +α2 −λ1 r(x)−λ2 x) 1 1 s+λ1 +λ2 +α2 −λ1 r(x)−λ2 B × exp dx dy, e2 (s + λ1 + λ2 + α2 − λ1 r(x) − λ2 x) − x µ y B D2 (s) =
1 e1 (s + α1 )][1 − B e2 (s + α2 + λ1 − λ1 r(1))] [1 − B n o e2 (s+α2 )+(s+λ1 −λ1 r(1))×[B e2 (s+α2 +λ1 −λ1 r(1))− B e2 (s+α2 )]D1 (s) , × 1− B
D3 (s) =
1 − (s + λ1 − λ1 r(1))D1 (s) , e2 (s + α2 + λ1 − λ1 r(1)) 1−B
e1 (s + λ1 + λ2 + α1 − λ1 z1 − λ2 z2 ) = 0 in the disk r(z2 ) is the unique root of equation z1 − B e2 (s+λ1 +λ2 +α2 −λ1 r(z2 )−λ2 z2 ) = 0 |z1 | ≤ 1, and ω(s) is the unique root of the equation z2 − B inside |z2 | = 1. Proof To find the reliability of the server, we let the failure states of the server be absorbing states, and we obtain a new system. In the new system, we use the same notations as in the previous section, then we can get the following set of equations: · ¸ 2 Z ∞ X d + λ1 + λ2 + jµ p00j (t) = pk0j (t, x)βk (x)dx, (4.2) dt 0 k=1
if i ≥ 1, ¸ ∂ ∂ + + λ1 + λ2 + α1 + β1 (x) p1ij (t, x) = λ1 p1,i−1,j (t, x) + λ2 p1,i,j−1 (t, x), ∂t ∂x · ¸ ∂ ∂ + + λ1 + λ2 + α2 + β2 (x) p2ij (t, x) = λ1 p2,i−1,j (t, x) + λ2 p2,i,j−1 (t, x), ∂t ∂x ·
p0ij (t) = 0,
(4.3) (4.4) (4.5)
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JINTING WANG
with the initial condition p000 (0) = 1 and boundary conditions: Z ∞ Z ∞ p1ij (t, 0) = δ0i λ1 p00j (t) + p1,i+1,j (t, x)β1 (x)dx + p2,i+1,j (t, x)β2 (x)dx, 0
(4.6)
0
p2ij (t, 0) = δ0i [λ2 p00j (t) + (j + 1)µp0,0,j+1 (t)].
(4.7)
By taking Laplace transforms of these equations, we obtain sp∗00j (s) − δ0j = −(λ1 + λ2 + jµ)p∗00j (s) +
2 Z X k=1
p∗0ij (s) = 0, sp∗1ij (s, x) +
sp∗2ij (s, x) +
∞
0
βk (x)p∗k0j (s, x)dx,
if i ≥ 1,
(4.9)
∂p∗1ij (s, x) = −(λ1 + λ2 + α1 + β1 (x))p∗1ij (s, x) ∂x +λ1 p∗1,i−1,j (s, x) + λ2 p∗1,i,j−1 (s, x), ∂p∗2ij (s, x) ∂x
(4.8)
(4.10)
= −(λ1 + λ2 + α2 + β2 (x))p∗2ij (s, x)
+λ1 p∗2,i−1,j (s, x) + λ2 p∗2,i,j−1 (s, x), ¸ 2 ·Z ∞ X ∗ ∗ ∗ p1ij (s, 0) = δ0i λ1 p00j (s) + pk,i+1,j (s, x)βk (x)dx , k=1
(4.11) (4.12)
0
p∗2ij (s, 0) = δ0i [λ2 p∗00j (s) + (j + 1)µp∗0,0,j+1 (s)]
(4.13)
Define the following generating functions: p∗00 (s, z2 ) =
∞ X
p∗00j (s)z2j ,
p∗1i (s, x, z2 ) =
j=0
∞ X
p∗1ij (s, x)z2j ,
p∗2i (s, x, z2 ) =
j=0
p∗1 (s, x, z1 , z2 ) =
∞ X ∞ X
p∗1ij (s, x)z1i z2j ,
∞ X
p∗2ij (s, x)z2j ,
j=0
p∗2 (s, x, z1 , z2 ) =
i=0 j=0
∞ ∞ X X
p∗1ij (s, x)z1i z2j .
i=0 j=0
Multiplying Equations (4.8) to (4.13) by z2j and summing over j, we obtain the following basic equations after some algebraic manipulations: 2
(s + λ1 +
λ2 )p∗00 (s, z2 )
∂p∗ (s, z2 ) X + − 1 = −zµ 00 ∂z
·Z 0
k=1
sp∗1i (s, x, z2 ) +
sp∗2i (s, x, z2 ) +
∞
¸ ,
βk (x)p∗k0 (s, x, z2 )dx
∂p∗1i (s, x, z2 ) = −(λ1 + λ2 + α1 + β1 (x))p∗1i (s, x, z2 ) ∂x +λ1 p∗1,i−1 (s, x, z2 ) + λ2 z2 p∗1,i (s, x, z2 ), ∂p∗2i (s, x, z2 ) ∂x
(4.15)
= −(λ1 + λ2 + α2 + β2 (x))p∗2i (s, x, z2 )
+λ1 p∗2,i−1 (s, x, z2 ) + λ2 z2 p∗2,i (s, x, z2 ), ¸ 2 ·Z ∞ X ∗ ∗ ∗ p1i (s, 0, z2 ) = δ0i λ1 p00 (s, z2 ) + pk,i+1 (s, x, z2 )βk (x)dx , k=1
p∗2i (s, 0, z2 )
(4.14)
(4.16) (4.17)
0
¸ · ∂p∗0,0 (s, z2 ) ∗ . = δ0i λ2 p00 (s, z2 ) + µ ∂z2
(4.18)
ON THE SINGLE SERVER RETRIAL QUEUE
313
Similarly, by multiplying Equations (4.15) to (4.18) by z1i and summing over i, we get ∂p∗1 (s, x, z1 , z2 ) ∂x = −(λ1 + λ2 + α1 + β1 (x))p∗1 (s, x, z1 , z2 ) + λ1 z1 p∗1 (s, x, z1 , z2 ) +λ2 z2 p∗1 (s, x, z1 , z2 ),
(4.19)
∂p∗2 (s, x, z1 , z2 ) ∂x = −(λ1 + λ2 + α2 + β2 (x))p∗2 (s, x, z1 , z2 ) + λ1 z1 p∗2 (s, x, z1 , z2 ) +λ2 z2 p∗2 (s, x, z1 , z2 ),
(4.20)
sp∗1 (s, x, z1 , z2 ) +
sp∗2 (s, x, z1 , z2 ) +
p∗1 (s, 0, z1 , z2 )
¸ 2 ·Z ∞ 1 X ∗ = + pk (s, x, z1 , z2 )βk (x)dx z1 0 k=1 ¸ 2 ·Z ∞ 1 X ∗ − pk,0 (s, x, z2 )βk (x)dx , z1 0 λ1 p∗00 (s, z2 )
(4.21)
k=1
p∗2 (s, 0, z1 , z2 ) = λ2 p∗00 (s, z2 ) + µ
∂p∗0,0 (s, z2 ) . ∂z2
(4.22)
From (4.19) and (4.20) we obtain p∗1 (s, x, z1 , z2 ) = p∗1 (s, 0, z1 , z2 ) exp{−(s + λ1 + λ2 + α1 − λ1 z1 − λ2 z2 )x}B 1 (x), (4.23) p∗2 (s, x, z1 , z2 ) = p∗2 (s, 0, z1 , z2 ) exp{−(s + λ1 + λ2 + α2 − λ1 z1 − λ2 z2 )x}B 2 (x), (4.24) which give e1 (s + λ1 + λ2 + α1 − λ1 z1 − λ2 z2 ) 1−B s + λ1 + λ2 + α1 − λ1 z1 − λ2 z2
(4.25)
e2 (s + λ1 + λ2 + α2 − λ1 z1 − λ2 z2 ) 1−B , s + λ1 + λ2 + α2 − λ1 z1 − λ2 z2
(4.26)
p∗1 (s, z1 , z2 ) = p∗1 (s, 0, z1 , z2 ) and p∗2 (s, z1 , z2 ) = p∗2 (s, 0, z1 , z2 )
respectively. Substituting the above results into (4.21), we get 2 i 1 Xh ∗ ek (s + λ1 + λ2 + αk − λ1 z1 − λ2 z2 ) pk (s, 0, z1 , z2 )B z1 k=1 ¸ ·Z 2 ∞ X 1 ∗ pk,0 (s, x, z2 )βk (x)dx . (4.27) − z1 0
p∗1 (s, 0, z1 , z2 ) = λ1 p∗00 (s, z2 ) +
k=1
We insert the expression of and get
P2 k=1
hR
∞ 0
i p∗k,0 (s, x, z2 )βk (x)dx obtained from (4.27) into (4.14)
314
JINTING WANG
(s + λ1 + λ2 − λ1 z1 )p∗00 (s, z2 ) + z2 µ
∂p∗00 (s, z2 ) ∂z2
e1 (s + λ1 + λ2 + α1 − λ1 z1 − λ2 z2 ))p∗1 (s, 0, z1 , z2 ) = 1 − (z1 − B e2 (s + λ1 + λ2 + α2 − λ1 z1 − λ2 z2 ), +p∗2 (s, 0, z1 , z2 )B
(4.28)
which gives e2 (s + λ1 + λ2 + α2 − λ1 z1 − λ2 z2 ) − z2 )] µ[B
∂p∗00 (s, z2 ) ∂z2
e2 (s + λ1 + λ2 + α2 − λ1 z1 − λ2 z2 )]p∗00 (s, z2 ) = [s + λ1 + λ2 − λ1 z1 − λ2 B e1 (s + λ1 + λ2 + α1 − λ1 z1 − λ2 z2 )]p∗1 (s, 0, z1 , z2 ). −1 + [z1 − B
(4.29)
e1 (s+λ1 +λ2 +α1 −λ1 z1 −λ2 z2 ). Similar to the derivation Consider the coefficient f (z1 ) = z1 − B by Falin and Templeton[4] , we can show that f (z1 ) = 0 has exactly one root z1 = r(z2 ) in the disk |z1 | ≤ 1. Replacing z1 = r(z2 ) in the above equation, we have e2 (s + λ1 + λ2 + α2 − λ1 r(z2 ) − λ2 z2 ) − z2 )] µ[B
∂p∗00 (s, z2 ) ∂z2
e2 (s + λ1 + λ2 + α2 = [s + λ1 + λ2 − λ1 r(z2 ) − λ2 B ∗ −λ1 r(z2 ) − λ2 z2 )]p00 (s, z2 ) − 1.
(4.30)
e2 (s + λ1 + λ2 + α2 − λ1 r(z2 ) − λ2 z2 ) − z2 ), and it has exactly Consider the coefficient f (z2 ) = B one root ω(s) in the interval [0, 1]. Similar to the derivation in [10], we get that if z 6= ω(s), p∗00 (s, z2 ) Z ω(s) = z2
1
e2 (s + λ1 + λ2 + α2 − λ1 r(y) − λ2 y)] − y] µ[B (
1 × exp µ
Z
z2
y
) e2 (s + λ1 + λ2 + α2 − λ1 r(x) − λ2 x) s + λ1 + λ2 + α2 − λ1 r(x) − λ2 B dx dy. e2 (s + λ1 + λ2 + α2 − λ1 r(x) − λ2 x) − x B
For z2 = ω(s), we have directly from (4.21) that p∗00 (s, ω(s)) =
1 . (1 − ω(s))[s + λ1 + λ2 − λ1 r(ω(s)) − λ2 ω(s)]
From (4.22) and (4.29), we get the expressions of p∗1 (s, 0, z1 , z2 ) and p∗2 (s, 0, z1 , z2 ). Then, R∗ (s) = p∗00 (s, 1) + p∗1 (s, 1, 1) + p∗2 (s, 1, 1) = p∗00 (s, 1) +
e2 (s + λ2 ) e1 (s + λ1 ) 1−B 1−B p∗1 (s, 0, 1, 1) + p∗2 (s, 0, 1, 1). s + λ1 s + λ2
By substitution, we obtain the formula (4.1). Corollary 4.4 The mean time to the first failure (MTTFF) of the server is given by MTTFF = D1 (0) +
e1 (λ1 ) e2 (λ2 ) 1−B 1−B D2 (0) + D3 (0). λ1 λ2
ON THE SINGLE SERVER RETRIAL QUEUE
Proof It is derived from (4.1) and the equation MTTFF=
R∞ 0
315
R(t)dt = R∗ (s)|s=0 .
References [1] T. Yang and J. G. C. Templeton, A survey on retrial queues, Queueing Systems, 1987, 2(3): 201–233. [2] G. I. Falin, A survey of retrial queues, Queueing Systems, 1990, 7(2): 127–167. [3] V. G. Kulkarni and H. M. Liang, Retrial queues revisited, in Frontiers in Queueing (ed. by J. H. Dshalalow), CRC Press, Boca Raton, FL, 1997. [4] G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman & Hall, London, 1997. [5] J. R. Artalejo, Accessible bibliography on retrial queues, Mathematical and Computer Modelling, 1999, 30(3–4): 1–6. [6] J. R. Artalejo, A classified bibliography of research on retrial queues: Progress in 1990–1999, TOP, 1999, 7(2): 187–211. [7] B. D. Choi and Y. Chang, Single server retrial queues with priority calls, Mathematical and Computer Modelling, 1999, 30(3–4): 7–32. [8] B. D. Choi and K. K. Park, The M/G/1 retrial queue with Bernoulli schedule, Queueing Systems, 1990, 7(2): 219–228. [9] G. I. Falin, J. R. Artalejo, and M. Martin, On the single retrial queue with priority customers, Queueing Systems, 1993, 14(3–4): 439–455. [10] J. Wang, J. Cao, and Q. Li, Reliability analysis of the retrial queue with server breakdowns and repairs, Queueing Systems, 2001, 38(4): 363–380. [11] A. Aissani, A retrial queue with redundancy and unreliable server, Queueing Systems, 1995, 17(3– 4): 431–449. [12] J. R. Artalejo, New results in retrial queueing systems with breakdown of the servers, Statistica Neerlandica, 1994, 48(1): 23–36. [13] V. G. Kulkarni and B. D. Choi, Retrial queues with server subject to breakdowns and repairs, Queueing Systems, 1990, 7(2): 191–208. [14] T. Yang and H. Li, The M/G/1 retrial queue with the server subject to starting failure, Queueing Systems, 1994, 16(1–2): 83–96. [15] V. I. Meykshan and I. G. Fidelman, The design of communications networks with bypass routings when there are repeated calls and lines with undetected breakdowns, Telecommunications and Radio Engineering, 1995, 49(7): 40–44. [16] J. Cao and K. Cheng, Analysis of M/G/1 queueing system with repairable service station, Acta Mathematicae Applicatae Sinica, 1982, 5(2): 113–127. [17] M. Martin and J. R. Artalejo, Analysis of an M/G/1 queue with two types of impatient units, Advances in Applied Probability, 1995, 27(3): 840–861.
