This paper deals with a discretetime bulkservice
Recently there has been a rapid increase in the literature on discretetime 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 [
The continuoustime infinite buffer single server batch service queue with multiple vacations has been analyzed by Choi and Han [
As to the research to queueing systems with batch service and working vacations (or vacation), Yu et al. [
This paper focuses on a discretetime batchservice infinite buffer
The rest of the paper is arranged as follows. In the next section, the model of the considered queueing system is described. In Section
We consider a discretetime bulkservice 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
The customers are served in batches of variable capacity, the maximum service capacity for the server being
Various time epochs in EAS.
We assume that interarrival times, service times, and working vacation times are mutually independent.
In addition, the service discipline is first in first out (FIFO). Let
Using the lexicographical sequence for the states, the onestep transition block matrix can be written as
We have
Assume that
If
According to the onestep transition probability matrix, we can see which is not
Let
Hence,
From (
Using
If
Define
If
In the steady state, the queue length
Let the random variable
In the steady state, the PGF of waiting time for an arriving customer is given by
Firstly, we define
(1) The server is on a normal busy period and
Under this condition, an arriving customer has to wait for one period of service for
We have
(2) An arriving customer finds the server is on vacation.
Let
(A) A vacation is going on whereas all of the arrived customers have been served. Then, an arriving customer has to wait for one period of service for
Let
(B) If a vacation is over and
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,
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
Queue length distributions at random slots for







0  0.5580  0  0  0.3906  0 
1  0.0363  0.3128  1  0.0603  0.3539 
2  0.0022  0.0679  2  0.0037  0.1501 
3 

0.0128  3 

0.0299 
4 

0.0023  4 

0.0055 
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Sum  1  Sum  1 
Queue length distributions at random slots for







0  0.6092  0  0  0.4265  0 
1  0.0394  0.263  1  0.0654  0.3313 
2  0.0024  0.0623  2  0.0041  0.1319 
3 

0.012  3 

0.0277 
4 

0.0022  4 

0.0052 
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Sum  1  Sum  1 
Queue length distributions at random slots for







0  0.6095  0  0  0.4266  0 
1  0.0394  0.2627  1  0.0655  0.3312 
2  0.0024  0.0622  2  0.0041  0.1319 
3 

0.012  3 

0.0277 
4 

0.0022  4 

0.0052 
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Sum  1  Sum  1 
Queue length distributions at random slots for







0  0.6095  0  0  0.4267  0 
1  0.0394  0.2627  1  0.0655  0.3312 
2  0.0024  0.0622  2  0.0041  0.1318 
3 

0.012  3 

0.0277 
4 

0.0022  4 

0.0052 
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Sum  1  Sum  1 
Effect of
Effect of
Effect of
A
The authors wish to thank the referee for his careful reading of the paper and for his helpful suggestion. Meantime, this work is supported by the National Natural Science Foundation of China (no. 71171138), and the Talent Introduction Foundation of Sichuan University of Science & Engineering (2012RC23).