A Single Server Queue with Coxian-2 Service and One-Phase Vacation (M/C-2/M/1 Queue)

In this paper, we study a single server queueing system with Coxian-2 service.  In Particular, we study M/C-2/M/1 queue with Coxian-2 service and exponential vacation. We assume that units (customers) arrive at the system one by one in a Poisson process and the server provides one-by-one service based on first in first out (FIFO) rule. We obtained the steady state queue size distributions in terms of the probability generating functions, the average number of customers and their average waiting time in the system as well as in the queue.


Introduction
In queueing theory we study situations where units of some kind arrive at a service facility for receiving service, some of the units having to wait for service, and go out after service. A queue or a waiting line develops when the service facility cannot deal with the number of units requiring service.
A system is generally defined as something that has an input, output and transformation process, which changes the input into the output. The study of a queuing system provides us with some characteristics that can be used to measure the performance of the system, like the proportion of time the service channel is idle, the proportion of time the service channel is busy and the average waiting time of a customer. Using these and similar measures one can predict what will happen if certain changes are made in the components of the system. For many queueing systems the queue discipline that is used is first in, first 1 with probability with probability 1 where 1 X and 2 X are independent random variables having exponential distribution with respective mean The probability density function of the Coxian-2 distribution random variable X is given by ( ) ( ) ( ) For more details see Tijms [1]. In recent years, vacation queues have been developed as an important area of queueing theory. In classical queueing theory it was assumed that the server is always available in the system. However, this is not true in many real life situations. In many queueing systems such as the large production systems, computer systems or communication networks, there may be a need to stop the system from time to time for routine maintenance or for overhauling. Recently many researchers including Crammer [2], Doshi [3], Keilson and Servi [4], Shanthikumar [5], Madan and Saleh [6] [7] and Madan, Abu-Dayyeh and Tayyan [8] have studied some such queueing systems with server vacations.
In this paper we study the M/C-2/M/1 queue with Poisson arrivals Coxian-2 service and exponential vacation. Whenever a customer takes a service, his service time is a random variable distributed as Coxian-2. Further, we assume that after every service the server may take a vacation of random length with probability p or may continue the next service with probability (1-p). Whenever the server takes a vacation, his vacation time is distributed exponentially. We have obtained time-dependent as well as steady state queue size distribution. In addition, for the steady state we find the mean queue size, the mean system size and the mean waiting time of a customer.

Assumptions, Definitions and Equations Governing the System
In this work we assume that 2) Phase-k service is exponential with mean service time 1 3) The server's vacation period has an exponential distribution with mean p: -Probability that the server takes a vacation after completion of service.
Then we have the following set of equations: In order to give a detailed reasoning needed to get the above equations, we shall explain how Equation (1) has been obtained. We connect the system probabilities at time t with those at time t + Δt by considering After rearranging the terms in the above equations and letting 0 t ∆ → we obtain the following set of differential equations: Open Journal of Applied Sciences

Time Dependent Solution
Assuming that initially there are no customers in the system and the server is idle, we have the following initial conditions: Now by taking Laplace transformation of Equations [(8-1)-(8-7)] and using (9) we get the following: Next, we define the following probability generating functions in terms of their where z is a dummy variable and 1 z ≤ .
Multiply Equation (10-1) by 1 n z + and sum over n = 1 to ∞, and multiply (10-2) by z, then add them together we get

Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.