Steady-state Analysis of the GI/M/1 Queue with Multiple Vacations and Set-up Time

In this paper, we consider a GI/M/1 queueing model with multiple vacations and set-up time. We derive the distribution and the stochastic decomposition of the steady-state queue length, meanwhile, we get the waiting time distributions.


Introduction
Vacation queues servers to stop the customers' service at some periods, and the time during which the service is interrupted is called the vacation time. Vacation queue research originated from Levy and Yechial, then many researchers on queuing theory deal with this fields. So far, the theory frame whose core is the stochastic decomposition is developed and vacation queues have been applied successfully to many fields, such as computer systems, communication networking, electronic and call centers. Details can be seen in the surveys of Doshi and the monographs of Tian. For GI/M/1 type queues with server vacations, Tian used the matrix geometric solution method to analyze and obtained the expressions of the rate matrix and proved the stochastic decomposition properties for queue length and waiting time in a GI/M/1 vacation model with multiple exponential vacations.

Description of the model
Consider a classical GI/M/1 queue, inter-arrival times are i.i.d.r.vs. Let ) (x A and ) ( * s a be the distribution function and L.S transform of the inter-arrival time A of customers. The mean inter-arrival time is λ ,Service times during service period, vacation times and set-up times are assumed to be exponentially distributed with rate µ ,θ , β , respectively. We assume that the service discipline is FCFS.
Suppose n τ be the arrival epoch of nth customers with 0 τ =0. Let be the number of the customers before the nth arrival. Define τ 0, the nth arrival occurs during a service period, 1, the nth arrival occurs during a set -up period, 2, the nth arrival occurs during a vacation period.
services complete during an inter-arrival time. Therefore, we have .
The transition matrix of ( , ) n n L J can be written as the Block-Jacobi matrix The matrix P is a GI/M/1 type matrix.

Steady-state queue length distribution
where 1 γ is the unique roots in the range 0 As we known, if 1 ρ < ,θ >0, the first equation has the unique root 11

It can be verify that B[R]has the left invariant vector
If 1 < ρ , the stationary probability distribution of ( ) 10 02 π π π π is given by the positive left invariant vector (3) and satisfies the normalizing condition 02 π + 1 ) )( , , ( Then, we get Finally, we obtain the theorem. ( 1)( Proof. The PGF of v L is as follows: Substituting the above equation into (4),we obtain the distribution of d L .
We can get means

Waiting time distribution
Let W and ) ( s W be the steady-state waiting time and its LST, respectively. Firstly, let 0 H , 1 H , 2 H be the probability that the server is in the service(set-up, vacation) period when a new customer arrives. We can compute When a customer arrives, if there are k customers and the server is in the service period, the waiting time equals k service times by the rate µ . Then, we have When a customer arrives, if there are k customers and the server is in the set-up period, the waiting time is the sum of the residual set-up time and k service times by the rate µ . Then, we have

Numerical examples
In the above analysis,we obtain the expected queue length in the steady state. The difference of parameters may influence the queue length. So, we present numerical examples to explain.