Probabilistic Decomposition Method on the Server Indices of an M ξ / G / 1 Vacation Queue

This paper develops a probabilistic decomposition method for an M/G/1 repairable queueing system with multiple vacations, in which the customers who arrive during server vacations enter the systemwith probability p. Such a novel method is used to analyze themain performance indices of the server, such as the unavailability and themean failure number during (0, t]. It is derived that the structures of server indices are two convolution equations. Further, comparisons with existing methods indicate that our method is effective and applicable for studying server performances in single-server M/G/1 vacation queues and their complex variants. Finally, a stochastic order and production system with a multipurpose production facility is numerically presented for illustrative purpose.


Introduction
There are some effective and convenient analytic methods for single-server queues with a repairable server or service station.For example, the Markov renewal process method is used to study an M/G/1 queueing system with repairable service station in [1], the geometric process method introduced by Lam is applied to analyze the lifetime behaviors and repair times of deteriorating service station in [2,3], and the matrix-geometric method is available for GI/M/1 and M/E  /1 repairable queues in [4,5].It is well known that the supplementary variable method posed by Cox [6] is very important in dealing with some Poisson input queues with a repairable server.Many researchers, such as Wang [7], Ke et al. [8], Liu et al. [9], and Cao [10], have utilized this method for lots of repairable single-server queueing systems.The above approaches were applied to analyze some queueing indices, such as queue size, waiting time, and their stochastic decompositions, and the performance measures of the server, such as the mean times to the first failure, unavailability and failure frequency.However, the common methods mentioned above usually become too complicated to be solved especially when dealing with some Poisson input bulk arrival queues with a repairable server and their complex vacation variants.
In this paper, based on the renewal process theory and Laplace and Laplace-Stieltjes transforms we develop a probabilistic decomposition method to analyze the performance measures of the repairable server for a single-server M  /G/1 queue with variable input rate and multiple vacations.Our method is completely different from the methods used in [1][2][3][4][5][6][7][8][9][10] and reveals that the structures of the server indices in Poisson input single-server bulk arrival vacation queues are two convolution equations.Our analytic idea is presented as follows: (1) with the definition of "generalized server busy period", we get the conditional probability that the time  is during the generalized server busy period; (2) according to this probability and our probabilistic decomposition method, we obtain the unavailability and average failure number of the server, which derive two convolution equations; (3) finally, by means of a special case, comparisons are made between our new method and supplementary variable method.Comparisons indicate that our method is more effective and applicable for Poisson input single-server bulk arrival queues with a repairable server and their complex vacation variants.
The rest of the paper is organized as follows.Sections 2 and 3 give the queue assumptions and preliminaries, respectively.In Section 4 a probabilistic decomposition method is developed to analyze main server indices.A special case

Assumptions
we consider an M  /G/1 vacation queueing system with variable input rate as follows.
(1) The interarrival times between batch customers, {  ,  ≥ 1}, are independent identically distributed (i.i.d) random variables with distribution function (3) The server takes multiple vacations when the system becomes empty.Let   be the server's the th vacation time.Assume that   ,  ≥ 1 are i.i.d random variables with distribution function (),  ≥ 0 and finite mean ().The customers who arrive during server vacations enter the system with probability  (0 <  < 1) or lose with probability 1 − .Upon returning from a vacation, the server immediately serves one by one when there is a waiting queue or leaves for another vacation when there is an empty queue.
(4) The server consists of  unreliable units; these units may possibly fail if and only if the server is serving a customer.Once a unit fails, the server breaks down and cannot continue to serve.The failed unit will be repaired immediately.After the repair is completed, the server resumes operating and continues to serve the customer whose service has not been finished yet.
The service time for a customer is cumulative.
(5) During the repair of a unit, the server cannot operate and the other units cannot fail.After repair, the unit is as good as new.The lifetime   of unit  of the server has an exponential distribution   () = 1− −   ,  ≥ 0, and its repair time   obeys an arbitrary distribution   (),  ≥ 0 with a mean repair time   ,  = 1, 2, . . ., .
(6) All random variables are mutually independent.At the initial time  = 0, the server begins to serve when the number of customers presented in the system (0) > 0, or the server is idle and waits for the first batch arrival when (0) = 0.
Remark 1. Assumption ( 6) is practical and reasonable.But it is later proved that the steady-state performance indices of server are independent of initial state (0) = ,  ≥ 0.

Preliminaries
Let  and  denote the lifetime and repair time of server, respectively; then for  ≥ 0, the distribution functions of  and  are given, respectively, by Thus, the mean repair time of server is given by Definition 3. The "service completion time of a customer" represents the time interval from the epoch when the service for a customer begins to the epoch when the service of this customer ends, which includes possible repair times of server due to its unit failures in the process of serving this customer.Denote by χ the service completion time of customer ; it is obvious that χ ,  ≥ 1, are i.i.d.random variables.
Lemma 4 (see [1]).Let G() = ( χ ≤ ),  ≥ 1, then where Definition 5.The "generalized server busy period" represents the time interval from the epoch when the service begins to the epoch when the system becomes empty, which also contains possible repair times of server due to its unit failures in the process of service.Let b denote the generalized server busy period initiated with one customer and its distribution function is B() with Laplace-Stieltjes transform b().Similar to the discussions in an M/G/1 queue with generalization vacations [11], the following lemma holds.
where ρ = (1 + ) denotes the traffic intensity of the considered queue.
Denote by b⟨⟩ the generalized server busy period initiated with  customers; then b⟨⟩ can be expressed as b⟨⟩ = ∑  =1 b , where b , 1 ≤  ≤ , are mutually independent with the same distribution function as b.Let B⟨⟩ () = ( b⟨⟩ ≤ ); then we can get B⟨⟩ () = B() (); that is, B⟨⟩ () is the -fold convolution of B().Definition 7. The "system idle period" represents the time interval from the epoch when the system becomes empty to the epoch when batch customers enter the system.

Performance Indices of the Server
In this section, we develop a probabilistic decomposition method to analyze main performance indices of the server in the considered queue, including the conditional probability that the time  is during the generalized server busy period, the unavailability and the average failure number during (0, ].Further, it is derived that the structures of server indices are two convolution equations.Finally, a special case is presented to validate our results and make comparisons between our method and supplementary variable method.

The Conditional Probability That the Time 𝑡 is during the Generalized Server Busy Period
Theorem 8.For  ≥ 0, let   () =  (the time  is during the generalized server busy period |(0) = ); then for R() > 0, Laplace transforms of   (),  ≥ 0 are and in steady state, for system traffic intensity ρ = (1 + ) and  ≥ 0, one has where b() is determined by Lemma 6. Proof.
In order to investigate the unavailability and the failure number during (0, ] of server, we introduce a classical -unit series repairable system [12].For  ≥ 0, let Φ () =  (the system is repaired at time ) ,  * () = ∫ (16) Lemma 9 (see [12]).If R() > 0, then where 4.2.The Unavailability of the Server.The unavailability of the server at time ; that is, the probability that the server is repaired at time .
Proof.(i) According to the queue assumptions, the server is repaired at time  if and only if the time  is during one generalized server busy period, and the server is repaired at time .Consequently, using the law of total probability and renewal process theory, we have the decomposition of Φ 0 () as follows: where   () = (0 ≤  <  ⟨⟩ , the server is repaired at time ),  ≥ 1.
(iii) Taking Laplace transforms of (20) and (21), respectively, and utilizing ( 22) and (23), we get Performing similar operations in the proof of Theorem 8 on (25) and (26), we can complete the proof by Theorem 8 and Lemma 9.
where () is determined by Lemma 9.
Remark 12 (a special example).If  = 1, ( = 1) = 1, and (  = 0) = 1,  ≥ 1, then our model becomes an M/G/1 repairable queue with an unreliable server [10], in which the From Table 3, we observe that the influence of  on four indices is completely opposite to that of .Table 4 reports the effects of unpredictable events arrival rate  on production facility indices.Here we assume that (, , , , ) = (0.2, 0.8, 0.5, 2, 0.8).As shown in Table 4, when  increases, all production facility indices increase monotonously.Furthermore, for  ≥ 0.32, the system becomes unstable and the production facility is always busy.The trends shown by Tables 1-4 are as expected.
From the analysis presented in Tables 1-4, it can be concluded that under the stability condition, that is ρ < 1, the performance indices of production facility are affected by batch order arrival, batch order entering probability, batch order size, and unpredictable events arrival.But as ρ ≥ 1, production facility indices are not affected by batch order arrival and batch order size, and production facility is always busy.In this case, the system is unstable.

Conclusions
In this paper, we develop a probabilistic decomposition method to analyze the performance measures of the repairable server in a single-server M  /G/1 queue with pentering discipline during server vacations.Our method is completely different from common methods used in [1][2][3][4][5][6][7][8][9][10] and reveals that the structures of server indices in Poisson input single-server bulk arrival vacation queues are two convolution equations.A special case and comparisons with supplementary variable method indicate that our method is effective and applicable for Poisson input bulk arrival vacation queues with a repairable server and their complex variants.Finally, a stochastic order and production system with a multipurpose production facility is numerically presented for illustrative purpose.In the future, the server performance indices of discrete time bulk arrival vacation queues will be our further work using similar probabilistic decomposition method.

Lemma 6 .
If R() > 0, then b() is the solution with smallest absolute value in  of the equation  = g( +  − ()), and