The Discrete-Time Bulk-Service Geo/Geo/1 Queue with Multiple Working Vacations

This paper deals with a discrete-time bulk-service Geo/Geo/1 queueing system with infinite buffer space and multiple working vacations. Considering an early arrival system, as soon as the server empties the system in a regular busy period, he leaves the system and takes a working vacation for a random duration at time n. The service times both in a working vacation and in a busy period and the vacation times are assumed to be geometrically distributed. By using embedded Markov chain approach and difference operator method, queue length of the whole system at random slots and the waiting time for an arriving customer are obtained. The queue length distributions of the outside observer’s observation epoch are investigated. Numerical experiment is performed to validate the analytical results.


Introduction
Recently there has been a rapid increase in the literature on discrete-time queueing system with working vacations.These queueing models have been studied extensively and applied to computer networks, communication systems, and manufacturing systems.In the classical queueing system with server vacations, the server stops working during vacation periods.Suppose, however, that a system can be staffed with a substitute server during the times the main server is taking vacations.The service rate of the substitute server is different from (and probably lower than) that of the main server.This is the notion of working vacations recently introduced by Servi and Finn [1].They studied an //1 queue with multiple working vacations (//1/).Their work is motivated by the analysis of a reconfigurable wavelength-division multiplexing (WDM) optical access network.In 2006, Wu and Takagi [2] generalized Servi and Finn's //1/ queue to an //1/ queue.Baba [3] extended Wu and Takagi's work to a renewal input //1 queue with working vacations and derived the steady-state system length distributions at an arrival and arbitrary epochs.The //1 queue with single and multiple working vacations have been discussed in Li and Tian [4] and Tian et al. [5], respectively.Chae et al. [6] studied the //1 queue and //1 queue with single working vacation (SWV).The discrete-time infinite buffer //1 queue with multiple working vacations and vacation interruption has been studied in Li et al. [7,8].The discrete-time finite buffer //1 queue with multiple working vacations has been discussed by Goswami and Mund [9].All the above studies on discrete-time single server queues have been carried out under the assumption that a server serves singly at a time.However, there are many instances where the servers are carried out in batches to enhance the performance of the system.Over the last several years the discrete-time single server queues in batch service without vacations have been studied in Gupta and Goswami [10], Chaudhry and Chang [11], Alfa and He [12], and Yi et al. [13].Lately, This type of queueing systems raise interest once more by many scholars such as Banerjee et al. [14,15], Claeys et al. [16,17].
The continuous-time infinite buffer single server batch service queue with multiple vacations has been analyzed by Choi and Han [18], Chang and Takine [19].The //1 queue

System Description
We consider a discrete-time bulk-service infinite buffer space queueing system with server multiple working vacations according to the rule of an early arrival system.Assume that the time axis is slotted into intervals of equal length with the length of a slot being unity, and it is marked as 0, 1, 2, . . ., , . .., a potential arrival occurs in the interval (,  + ) and potential batch departures occur in ( − , ).In the meantime, the interarrival times  of customers are independent and geometrically distributed with probability mass function (p.m.f.) { = } =  −1 ,  ≥ 1,  = 1 − .The customers are served in batches of variable capacity, the maximum service capacity for the server being  ( ≥ 1).Service times   during normal busy period and service times   during a working vacation are assumed independent and geometrically distributed with p.m.f.{  = } =    −1  ,  ≥ 1,   = 1 −   and p.m.f.{  = } =    −1  ,  ≥ 1,   = 1 −   , respectively.A new arriving customer cannot go into the queue being served immediately in spite of the working vacation period and the normal busy period.The server leaves the system and takes a working vacation at epoch  as soon as system becomes empty in regular period.The working vacation time follows a geometric distribution with parameter  (0 <  < 1) and its p.m.f. is { = } =  −1 ,  ≥ 1,  = 1−.If there are some customers being served after the server finishes a working vacation, the service interrupted at the end of a vacation is lost and it is restarted with service rate   at the beginning of the following service period, which means that the regular busy period starts.The various time epochs at which events occur are depicted in Figure 1.
We assume that interarrival times, service times, and working vacation times are mutually independent.
We have

The Stationary Queue Size Distributions at Random Slots
Assume that (, ) is the stationary limit of {  ,   }, and its distribution is denoted as  , = lim  → ∞ {  = ,   = } = { = ,  = }, (, ) ∈ Ω, we have the following theorem. where Proof.According to the one-step transition probability matrix, we can see which is not QBD process, the method of matrix-geometric solution is invalid.Based on the stationary equations obtained directly by stochastic balance, we have Then, we obtain the following equations: According to the characteristic of difference equations, let  +,0 =    ,0 [24],  ∈ ;  = 0, 1, 2, . .., where  denotes the difference operator for the difference equation, substituting it into (10), ( 10) can be written as The auxiliary equation is given by Let () =    +1 +    +   +  and () = −.Using Rouché's theorem, it can be shown that there is only one zero real root falls in the unit circle (Note: the root must be real root; otherwise, there are two roots at least fall in the unit circle, because the imaginary roots of an equation appear in pairs.)We denote this root by  (0 <  < 1) and the other  roots by   , |  | ≥ 1 ( = 1, 2, 3, . . ., ).So  satisfies () + () = 0. Therefore, the solution of ( 10) can be written as Since   ( = 1, 2, 3, . . ., ) = 0 (otherwise, the probability  ,0 tends to ∞ when  tends to ∞), we get  ,0 =  0   ( ≥ 1).
Corollary 3. Define  as the state of system at random slots, the steady-state probability of this system at random slots can be written as and the average queue length is Proof.In the steady state, the queue length  at random slots has marginal distribution as Using () = { = 0} + ∑ ∞ =1 { = }  , we can obtain (28) easily; furthermore, taking derivation to () and let  = 1, we can get (29).

The Waiting Time Distribution
Let the random variable   be the total waiting time of an arriving customer in the queue,   represents the number the customers in the system.Assume that an arriving customer finds  customers in the system, the conditional distribution law that he waits for  slots is subject to   () = {  = /  = },  = 0, 1, 2, . . .,  = 0, 1, 2, . .., and PGF is   () = ∑ ∞ =0   ()  .In the steady state, the waiting time with finite mean   has PGF   () = ∑ ∞ =0     (),  = 0, 1.

Theorem 5. In the steady state, the PGF of waiting time for an arriving customer is given by
and the average waiting time as Proof.Firstly, we define ⌊⌋ as a greatest integer function (floor), which returns the greatest integer less than or equal   (40) The PGF of the waiting time can be given by Adding ( 33)-(41), we can get (31), using (  ()/) | =1 , we can obtain (32).

Outside Observer's Observation Epoch Distributions
For an early arrive system, since an outside observer's observation epoch falls in the time interval after a potential arrival and before a potential batch departure, let, π,0 , π,1 be  ( ≥ 0) customers in the system and the server is on vacation (including the servicing customers),  customers in the system and the server is in regular busy period (including the servicing customers).Through observing the relationship between random slot  and the outside observer's observation epoch ( * ), we have π0,0 =  0,0 , π,0 =  ,0 +  −1,0 , ( ≥ 1) , π1,1 =  1,1 +  0,0 +

Numerical Results and the Sensitivity Analysis of this System
In this section, we present some numerical results in selfexplanatory tables and graphs for queue length distributions at random slots and all the numerical results have been obtained using the results derived in this paper.We observe that  ,0 and π,0 monotonically increase whereas  ,1 and π,1 monotonically decrease as  increases in Tables 1-4.This situation continues until  is equal to some constant; all data will tend to be a steady state.The above description is consistent with actual situation.In the meantime, () and (  ) monotonically decease as  increases.In Figures 2 and 3, fixing  = 10,  = 0.3,   = 0.5, and  = 0.3, 0.5, 0.7, we have plotted the effect of various vacation service rates on the average queue length and the average waiting time, respectively.We observe that the average queue length and the average waiting time decrease as the vacation service rate increases.In Figure 4, fixing  = 0.3,   = 0.4,   = 0.6, and  = 0.7, the steady-state average queue length equals 0.2597 from  = 4 on and the steadystate average waiting time equals 0.8421 from  = 5 on.They do not change as the batch size increases.

Conclusions
A / [] /1/ queueing system has been investigated.Assume that the server takes a working vacation after emptying the system in regular busy period.By using embedded Markov chain approach and the method of nonhomogeneous and homogeneous difference operator, the number of customers of the whole system at random slots has been discussed.This is different from general batch service queue literatures (excluding customers being served).The waiting time for an arriving customer and numerical results are obtained.In the future, further study such as  [] / [] /1/ queue will be the research topic using similar idea and method.

Figure 2 :
Figure 2: Effect of   on the average queue length.

Figure 3 :
Figure 3: Effect of   on the average waiting time.

Figure 4 :
Figure 4: Effect of  on the average queue length and the average waiting time.