An M/g/1 Queue with Two-stage Heterogeneous Service and Single Working Vacation

distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We study an M/G/1 queue with two stages of heterogeneous service and single working vacation.Using supplementary variable approach we derive the probability generating function for the number of customers and the average number of customers in the system. Some special cases of interest are discussed.


Introduction
The M/G/1 queue with vacation time has been studied by number of authors.To mention a few references we would name Cooper (1981), Levy et.al. (1975), Yukata Baba (1986), Teghem (1986) and Doshi (1990).Recently a class of semi-vacation policies has been introduced by Servi and Finn.Such a vacation is called working vacation(WV).The server works at a lower rate rather than completely stops service during a vacation.Servi and Finn (2002) studied an M/M/1 queue with multiple working vacation and obtained the probability generating function for the number of customers in the system and the waiting time distribution.Some other notable works were done by Wu and Takagi (2006), Tian et.al. (2008) and Afthab Begum (2011).
In many queueing situations all arriving customers require the main service and only some of them may require the subsidiary service provided by the server.Madan (2000) investigated an M/G/1 queueing system with second optional service.The single server vacation queueing models with second optional service were analyzed by many authors including Kalyanaraman et.al. (2008) and Thangaraj (2010).
In this paper we study an M/G/1 queue with Two-stage Heterogeneous service and single working vacation(SWV).The organization of the paper is as follows.In section 2 we described the model.In section 3 we obtained the steady state probability generating function.In section 4 the performance measures are obtained and in section 5 particular cases have been discussed.

The Model description
We consider a single server queueing system in which customers arrive according to a Poisson process with mean arrival rate λ(> 0).The service discipline is FCFS.Each arriving customer undergoes the first essential stage (FES) of service which has general distribution with the distribution function S b 1 (x), the probability density function s b 1 (x) and the Laplace- where S b 1 is the service time of the first stage service.After the completion of FES of service the customer may opt for the second optional stage (SOS) of service with probability p or the customer may leave the system with out taking the SOS of service with probability q(p + q = 1).
The SOS of service also follows the general distribution with the distribution function S b 2 (x), the probability density function s b 2 (x) and S * b 2 (θ) be the LST of S b 2 (x), where S b 2 is the service time of the second stage service.
Whenever the system becomes empty at a service completion instant the server starts working vacation and the duration of the vacation time follows an exponential distribution with rate η.At a vacation completion instant, if there are customers in the system the server will start a new busy period.
Otherwise he waits until a customer arrive.This type of vacation policy is called single working vacation.During the working vacation period the server also provides two stages of service.The first essential stage of service time S v 1 of a typical customer follows a general distribution with the distribution function S v 1 (x)[s v 1 (x) the probability density function and S * v 1 (θ), the LST] and the SOS of service time S v 2 also follows a general distribution with the distribution function S v 2 (x)[s v 2 (x) the probability density function and Further, it is noted that the service interrupted at the end of a vacation is lost and it is restarted with a different distribution at the beginning of the following service period.Inter arrival times, service times and working vacation times are mutually independent of each other.

The System Size Distribution at a Random Epoch
The system size distribution at an arbitrary time will be treated by the supplementary variable technique.That is, from the joint distribution of the queue length and the remaining service time of the customer in service if the server is busy, or the remaining service time of the customer if the server is on WV.
Assuming that the system is in steady state condition.Let us define the following random variables.N(t)the system size at time t.S 0 b 1 (t)-the remaining service time for the FES of service in not WV period.
S 0 b 2 (t)-the remaining service time for the SOS of service in not WV period.S 0 v 1 (t)-the remaining service time for the FES of service in WV period.S 0 v 2 (t)-the remaining service time for the SOS of service in WV period.
We define the following limiting probabilities: By considering the steady state, we have the following system of the differential difference equations.
We define Laplace Stieltjes transforms and probability generating functions as follows, Taking the LST of (2) to ( 5) and ( 7) to (10) we have y)dy ( 15) z n times (12) summing over n from 2 to ∞ is added up with z times (11) we z n times ( 14) summing over n from 2 to ∞ is added up with z times (13) we Inserting θ = (λ − λz + η) in ( 19) and (20) we get The denominator of the above equation has a unique root z 1 in (0, 1).Therefore 22) and ( 23) in ( 19) and (20) and putting θ = 0, we have z n times ( 16) summing over n from 2 to ∞ is added up with z times (15) we z n times (18) summing over n from 2 to ∞ is added up with z times (17) we Inserting θ = (λ−λz) in ( 26) and ( 27) and also substituting qP 1,1 (0 and Substituting ( 28) and ( 29) in ( 26) and ( 27) and inserting θ = 0, we get where as the probability generating function for the number of customers in the system when the server is on not WV period and as the probability generating function for the number of customers in the system when the server is on WV period then as the probability generating function for the number of customers in the system.We shall now use the normalizing condition P (1) = 1 to determine the only unknown Q 0 , which appears in (36).Substituting z = 1 in (36) and using L'hospital's rule we obtain on simplification where ) and E(S b 2 ) are the mean service times of stage 1 and stage 2 respectively.From (37) we obtain the system stability condition,

Mean System Length
Let L v and L b denote the mean system size during the working vacation and not working vacation period respectively.Then (D(1)) 2  where and D 2 (z) are given respectively in equations ( 32) and ( 33) where