Acta Mathematicae Applicatae Sinica, English Series Vol. 25, No. 4 (2009) 675–684 DOI: 10.1007/s10255-008-8823-1 http://www.ApplMath.com.cn
Acta Mathemaca Applicatae Sinica, English Series The Editorial Office of AMAS & Springer-Verlag 2009
A Single-server Discrete-time Retrial G-queue with Server Breakdowns and Repairs Jin-ting Wang1 , Peng Zhang2 1 Department 2 Qingdao
of Mathematics, Beijing Jiaotong University, Beijing 100044, China (E-mail:
[email protected])
Branch, The Bank of Communication, Qingdao 266001, China (E-mail:
[email protected])
Abstract This paper concerns a discrete-time Geo/Geo/1 retrial queue with both positive and negative customers where the server is subject to breakdowns and repairs due to negative arrivals. The arrival of a negative customer causes one positive customer to be killed if any is present, and simultaneously breaks the server down. The server is sent to repair immediately and after repair it is as good as new. The negative customer also causes the server breakdown if the server is found idle, but has no effect on the system if the server is under repair. We analyze the Markov chain underlying the queueing system and obtain its ergodicity condition. The generating function of the number of customers in the orbit and in the system are also obtained, along with the marginal distributions of the orbit size when the server is idle, busy or down. Finally, we present some numerical examples to illustrate the influence of the parameters on several performance characteristics of the system. Keywords
Discrete-time retrial queue; G-queue; Markov chain; unreliable server
2000 MR Subject Classification
1
60K25; (90B22)
Introduction
G-networks were first introduced by Gelenbe in 1989 (see [15]) with a view to modelling neural networks and they are characterized by the following feature: in contrast with the normal positive customers, negative customers arriving to a non-empty queue remove an amount of work from the queue. The G-networks systems have a rich and wide range of real applications in areas such as computers, communications and manufacturing (see [13, 14, 16, 19–25]). For a recent systematic account of the main methods and results on this topic, we refer to Gelenbe[17,18] , Artalejo[2] and references therein. Recently, the analysis of G-queues has been extended to the discrete-time situation in view of their applicability in the study of practical problems arise from telecommunication networks and computer networks [9, 11]. In [9], the authors consider a discrete-time Geo/Geo/1 queue with negative customers and disasters, in which the RCH (removal of customers from the head of the queue) killing discipline, RCE (removal of customers from the end of the queue) killing discipline and disaster discipline are studied. In a subsequent paper [11], the same authors analyze a discrete-time single-server queue with geometrical arrivals of both positive and negative customers. The RCE-inimmune servicing killing policy and RCE-immune servicing killing policy are investigated regarding a customer in service can and cannot be killed by a negative customer. The G-queues studied in the above papers deal with systems in which the customer at Manuscript received March 5, 2009. Revised April 15, 2009. Supported by the National Natural Science Foundation of China (No. 10871020).
J.T. Wang, P. Zhang
676
the head of the queue is able to occupy the server just at the service completion epoch of the customer being served. However, there is another type of queueing situation in which an arriving customer who finds the server unavailable upon arrival leaves the service area and, after some time, repeats his demand. Between trials a customer is said to be in orbit. These models with repeated attempts are called ‘retrial queues’ and they can be considered a type of network with reservicing after blocking which is widely used to model many problems in telephone switching systems, telecommunication networks and computer networks. With the recent advancements in mobile communications the issue of retrials is becoming more important. In the retrial queueing literature, Artalejo and G´ omez-Corral[3−8] have investigated single server queues operating under the simultaneous presence of negative arrivals and repeated attempts. However, these works focus on the continuous cases. It is well known that in the field of telecommunications modern systems are more digital than analogue, hence the studies of the modern communication systems are based more on discrete-time analysis. But, as far as we know, there are no results for the discrete retrial G-queueing systems although the analysis of discrete-time retrial queueing models has received considerable attention in the literature over the past few years (see [1, 10, 12, 29–31] and references therein). The present paper studies a discrete-time single-server retrial queue with both positive and negative customers. An example of a queueing system with both positive and negative flows of arrivals is computer networks with virus infection. When there are no viruses, computer networks have been modeled and analyzed using conventional queueing networks with or without retrials. The customers represent the execution of algorithmic and logic operations, and the nodes represent CPUs, I/O devices, etc. When a virus enters a node, one or more files may be infected, and the system manager may have to go through a number of backups to recover the infected files. In some cases, they may not be recoverable. A virus may originate from outside the network, e.g., through a floppy disk, or may come from another node in the network, e.g., by an electronic mail. The recover time of the infected files can be regarded as repair time of the server (node) due to an arrival of negative customer (virus). The primary contribution of this work is extending the retrial queueing theory on negative arrivals to discrete-time retrial G-queueing systems. Another contribution of this paper is that the reliability modelling using G-queues is extended to discrete-time situation for the first time. It is noted that in a recent work [26] Harrison et al. present related reliability modelling work in continuous-time case, and later, Wang et al.[28] further investigate an M/G/1 retrial G-queue with unreliable server in continuous time regime.
m
-
m+
(m+1)
m
(m+1)+
-
(m+1)
Potential departure epoch
End of the repairs
Potential arrival epoch
Beginning of the repairs
m+: Epoch after a potential arrival
Retrial epoch
m - : Epoch prior to a potential departure
Figure 1. Various time epochs in an early arrival system (EAS)
2
Model Description
We consider a Geo/Geo/1 retrial G-queue with unreliable server where the time axis is segmented into a sequence of equal intervals (called slot) and the time axis be marker by 0, 1, · · · , m, · · ·. It is assumed that all queueing activities (arrivals, departures, retrials, breakdowns and
A Single-server Discrete-time Retrial G-queue with Server Breakdowns and Repairs
677
repairs) occur at the slot boundaries. Different from continuous-time queues, the probability of an arrival and a departure and other queueing activities occurring concurrently may be not zero in discrete-time queues anymore. So it is necessary to specify the order in which the arrivals and departures take place in case of simultaneity. For mathematical clarity, we assume that the departures and the end of the repairs occur in the interval (m− , m), and the arrivals, the retrials and the beginning of the repairs occur in the interval (m, m+ ); that is, the arrivals, the retrials and the beginning of the repairs occur at the moment immediately after the slot boundaries, and the departures and the end of the repairs occurs at the moment immediately before the slot boundaries. In the literature it is called “early arrival system” (EAS). The model under consideration can be viewed through a self-explanatory figure (see Fig. 1). Readers are referred to Hunter[27] for more details. New customers arrive from outside the system according to a geometric arrival process with rate p. The arriving customers have two types: positive customers (with probability q + ) and negative customer (with probability q − ). Positive customer (external or repeated) receives the service immediately when the server is found free. If the server is found busy or down, the customer must join the orbit and return after random amount of time (in slots). Each customer in the orbit generates independently a Bernoulli stream with rate r = 1 − r, where r is the probability that a repeated customer does not make a retrial in a slot. The arrival of a negative customer causes one positive customer to be killed if any is present, and simultaneously breaks the server down. The failed server is sent to repair immediately and after repair it is assumed as good as new. The negative customer also causes the server breakdown if the server is found idle, but has no effect on the system if the server is under repair. The repair times are independent and geometrically distributed with parameter α = 1−α, where α is the probability that the repair process does not complete in a slot. The service times are independent and geometrically distributed with probability s = 1 − s, where s is the probability that a customer does not conclude his service in a slot. After service completion, the customer leaves the system immediately forever. Finally, various stochastic processes involved in the system are assumed to be independent of each other. To avoid trivial cases, we suppose 0 < p < 1, 0 ≤ r < 1, 0 ≤ q + ≤ 1 and 0 < s ≤ 1.
3
The Markov Chain
At time m+ , the system can be described by the process Ym = (Cm , Nm ), where Cm denotes the state of the server (0, 1 or 2 according to whether the server is free, busy or down) and Nm , the number of repeated customers in the retrial orbit. It can be shown that {Ym , m ∈ N} is the Markov chain of our queueing system, whose state space is Ω = {(i, k) : i = 0, 1, 2, k ≥ 0}. Let π0,k = lim P [Cm = 0, Nm = k],
k ≥ 0,
π1,k = lim P [Cm = 1, Nm = k],
k ≥ 0,
π2,k = lim P [Cm = 2, Nm = k],
k ≥ 0,
m→∞ m→∞ m→∞
be the stationary distributions of the Markov chain {Ym , m ∈ N}. Our object is to find the stationary distributions. By routine method, the Kolmogorov equations for the stationary distribution of our system are:
J.T. Wang, P. Zhang
678
π0,k = prk π0,k + psrk π1,k + pαrk π2,k , +
+
(3.1) +
π1,k = pq π0,k + (ps + pq s)π1,k + pq απ2,k + p(1 − r k+1
π2,k
k+1
)π0,k+1
k+1
+ ps(1 − r )π1,k+1 + pα(1 − r )π2,k+1 + (1 − δ0,k )pq + sπ1,k−1 , = pq − π0,k + pq − π1,k + (pα + pq − )π2,k + (1 − δ0,k )pq + απ2,k−1 ,
(3.2) (3.3)
where p = 1 − p, and δi,j denotes Kronecker’s delta. The normalization condition is given by ∞ ∞ ∞ π0,k + π1,k + π2,k = 1. k=0
k=0
k=0
In order to solve Eqs.(3.1)–(3.3), we introduce the following generating functions: ∞ ∞ ∞ π0,k z k , ϕ1 (z) = π1,k z k , ϕ2 (z) = π2,k z k . ϕ0 (z) = k=0
k=0
k=0
Multiplying Eqs.(3.1)–(3.3) by z k and summing over k, we get ϕ0 (z) = pϕ0 (rz) + psϕ1 (rz) + pαϕ2 (rz),
(3.4)
p p ϕ1 (z) = pq + ϕ0 (z) + (ps + pq + s)ϕ1 (z) + pq + αϕ2 (z) + ϕ0 (z) − ϕ0 (rz) z z ps ps pα pα + ϕ2 (z) − ϕ2 (rz) + pq szϕ1 (z), + ϕ1 (z) − ϕ1 (rz) + z z z z ϕ2 (z) = pq − ϕ0 (z) + pq − ϕ1 (z) + (pα + pq − )ϕ2 (z) + pq + αzϕ2 (z).
(3.5)
By substituting (3.4) into (3.5) we obtain p ps ϕ0 (z) + ps + pq + s + + pq + sz ϕ1 (z) ϕ1 (z) = pq + + z z pα 1 + ϕ2 (z) − ϕ0 (z). + pq α + z z
(3.6)
(3.7)
Solving the system (3.6) and (3.7), and after simplifying the expressions we find the following generating function: (1 − z)pq + (−pq + αz + pα + p) ϕ0 (z) ϕ1 (z) = , (3.8) Θ(z) (1 − z)pq − (−pq + sz + ps + p) ϕ0 (z) , (3.9) ϕ2 (z) = Θ(z) where Θ(z) ≡ pq − (pq + αz + pα) − (psz + pq + sz + pq + sz 2 + ps − z)(pα + pq − + pq + αz − 1). Before we solve these equations, we give two lemmas for later use. Lemma 3.1.
(1) The inequality Θ(z) > 0 holds for 0 ≤ z < 1 if and only if pq + pq − < 1 − ; s + spq − α + αpq −
(2) Under the above condition there exists 1−z 1 = lim . − + z→1 Θ(z) (pq + ps − pq s)(pα + pq + α) − pq + pq − Proof.
(1) By definition, Θ(z) can be expressed as a quadratic function of z: Θ(z) =(1 − z){pq + spq + αz 2 − [pq + α(ps + pq − ) + pq + s(pα + pq + )]z + (pq − pα + pspα + pspq + )}.
A Single-server Discrete-time Retrial G-queue with Server Breakdowns and Repairs
679
Now, letting f (z) ≡ pq + spq + αz 2 − [pq + α(ps + pq − ) + pq + s(pα + pq + )]z + (pq − pα + pspα + pspq + ), then it is readily seen that f (z) > 0 for 0 ≤ z < 1 if and only if pq + pq − < 1 − . s + spq − α + αpq − +
−
pq pq To prove the sufficiency of Lemma 3.1(1), we start with the condition s+spq − < 1 − α+αpq − . It is not difficult to see that pq + pq − (pq − + ps − pq + s)(pα + pq + α) − pq + pq − 1− > 0, (3.10) − = − − s + spq α + αpq (p + ps − pq + s)(pq − + pq + α + pα)
pq + < s + spq − .
and thus,
(3.11)
Setting z = 0, 1 in f (z) respectively, we get f (0) = pq − pα + pspα + pspq + > 0, f (1) = (pq − + ps − pq + s)(pα + pq + α) − pq + pq − > 0. On the other hand, the axis of symmetry of the function is given by z=
pq + α(ps + pq − ) + pq + s(pα + pq + ) > 1, 2pq + spq + α
where the last inequality is derived from the following expression with the help of Eq. (3.11): pq + α(ps + pq − ) + pq + s(pα + pq + ) − 2pq + spq + α =pq + α(s + spq − − pq + ) + pq + s(pα + pq + α) > 0. So, f (z) is monotone on [0, 1). Since f (0) > 0, f (1) > 0, it is readily seen that Θ(z) > 0 by virtue of the property of the quadratic function. To prove the necessity, we start from the inequality Θ(z) > 0, or, equivalently, f (z) > 0. We set z = 1 and get f (1) = (pq − + ps − pq + s)(pα + pq + α) − pq + pq − . Now suppose that f (1) ≤ 0, since f (z) is a continuous function, there exists ε > 0, such that f (1 − ε) ≤ 0. But, since 0 < 1 − ε < 1, we have f (1 − ε) > 0 by assumption. That is contradiction, therefore we have that f (1) > 0. Furthermore, we have pq + pq − (pq − + ps − pq + s)(pα + pq + α) − pq + pq − 1− − = s + spq − α + αpq − (p + ps − pq + s)(pq − + pq + α + pα) f (1) > 0, = (p + ps − pq + s)(pq − + pq + α + pα) that is,
pq + pq − 0 is equivalent to s+spq (2) From Lemma 3.1(1), we know that Θ(z) − < 1 − α+αpq − for 0 ≤ z < 1. By the L’Hospital’s rule one can derive the desired result under the above condition. 2 Using the previous lemma, the auxiliary generating functions ϕ1 (z), ϕ2 (z) are defined for 0 ≤ z < 1, and in z = 1 are extended by continuity if and only if pq + pq −
