M/G/1 Multistage Queue with Stand-by Server during Main Server’s Interruptions

Received Sep 24, 2017 Revised Nov 10, 2017 Accepted Dec 11, 2017 This paper investigates a multistage batch arrival queue with different server interruptions and a second server replaces the main server during the interruptions. The different server interruptions are assumed to be: multiple vacation, extended vacation, breakdown with delay time and server under two phases of repair. Customers are assumed to arrive in batches according to Poisson process and a single server provides service to the customers. When the main server is inactive due to the interruptions, stand-by server provide service to the arrivals. In addition, customers may renege during server breakdown or during server vacation due to impatience. Transient solution and the corresponding steady state solution is derived using supplementary variable technique.


INTRODUCTION
Many authors have studied queueing models by importing many aspects on them such as server vacation, breakdown, repair, reneging, balking etc.
A batch arrival queueing model with different vacation policies is studied by numerous authors including R. Vimaladevi [1], G.Ayyappan & S.Shyamala [2] and Ke.J.C, Huang.H.I and Chu.Y.K [3]. G. Ayyappan and S. Shyamala [4] obtained transient state and steady state of batch arrival queue subject to random breakdowns and Bernoulli schedule server vacation with second optional repair. Khalaf [5] studied the single stage queueing model with single original and extended vacation having breakdown with delay time and the system equipped with a stand-by-server works during the main server stops. Monita Baruah, K.C.Madan and Tillal Eldabi [6] studied a two stage batch arrival queue with reneging during vacation and breakdown periods. Batch arrival retrial queue with multi optional repair is discussed by D.Sumitha and K.Udaya Chandrika [7]. Multistage Batch arrival queue subject to different vacation policy with two phases of repair is discussed in C.Yuvarani and C.Vijayalakshmi [8].
In this work, we consider M[x]/G/1 queue with 'N' stages of services under different vacation policy and extended vacation subject to system breakdown with delay time and two phase of repairs. In addition, we assume that the customers may renege during breakdown or vacation period due to impatience.
This paper is organized as follows. The assumptions of our model are given in section 2. Definitions and Equations governing the system are given in section 3

MODEL ASSUMPTIONS
We assume the following to describe the queueing model of our study. a) The mean arrival rate of customers in batches is λ and they are served one by one on a first come -first served basis. The first order probability that a batch of i customers arrives at the system during a short interval of time ( . b) A single server provides 'N' stages of services for each customer, with the service times having general distribution. Let B i (x) and b i (x) (i =1, 2, 3,…,N) be the distribution and the density function of i stage service respectively. Let dx x i ) ( µ be the conditional probability density of service completion during the interval (x; x + dx], given that the elapsed time is x, so that After service completion of a customer, the server may remain in the system to serve the next customer with probability 0 β or he may proceed on j th vacation scheme with probability j β Once the original vacation gets over, the server has an option of taking an extended vacation with probability p or he may rejoin the system immediately with probability 1-p. The server's extended vacation time follows a general (arbitrary) distribution with distribution function W j (x) and density function w j (x). Let dx x j ) ( θ be the conditional probability of a completion of a extended vacation during the interval (x; x + dx] given that the elapsed extended vacation time is x, so that Reneging is assumed to follow exponential distribution with parameter η. Thus ηdt is the probability that a customer can renege during a short interval of time (t; t + dt]. f) Once the server breakdown, the repair do not start immediately. There is a delay time to start the repairs.
The delay time follows general distribution with distribution and density function H(x) and h(x) respectively. Let dx x) ( φ be the conditional probability of a completion of a delay time during the interval (x; x + dx] given that the elapsed delay time is x, so that The server breakdown is assumed to occur according to a poisson stream with mean breakdown rate α > 0. h) When the server breakdown the repair process may start any time. First the server sent for first essential repair (FER). After the completion of FER, the server may opt for the second optional repair (SOR) with The stand-by server provide service to the customers when the main server is on vacation, extended vacation, waiting for repair to start or under repair until the main server returns. The stand-by service time is assumed to follow exponential distribution with parameter δ.

DEFINITIONS AND EQUATIONS GOVERNING THE SYSTEM
We define denotes the probability that there are 'n' (n ≥ 0) customers in the queue excluding the one in service at time t and the server is active providing i th stage (i =1, 2, 3,…,N) of service with the elapsed service time for this customer is x and its corresponding probability irrespective of value of x is denoted by denotes the probability that at time t, the server is on j th vacation (j =1, 2, 3,…,M) and there are 'n' (n ≥ 1) customers waiting in the queue for service and its corresponding probability irrespective of value of x is denoted by ) ( (iii) R n (1) (x,t) denotes the probability that at time t, the server is inactive due to breakdown and the system is under FER while there are 'n' (n ≥ 0) customers in the queue and its corresponding probability irrespective of value of x is denoted by R n (1) (t). (iv) R n (2) (t) denotes the probability that at time t, the server is inactive due to breakdown and the system is under SOR while there are 'n' (n ≥ 0) customers in the queue and its corresponding probability irrespective of value of x is denoted by R n (2) (t). (v) Q(t) = Probability that at time t, there are no customers in the queue and the server is idle but available in the system.
denotes the probability that at time t, the server is on j th extended vacation (j =1, 2, 3,…,M) and there are 'n' (n ≥ 1) customers waiting in the queue for service and its corresponding probability irrespective of value of x is denoted by (vii) D n (x,t) denotes the probability that there are 'n' (n ≥ 0) customers waiting in the queue for service at time t, and the server is waiting for repair to start with elapsed delay time x and its corresponding probability irrespective of value of x is denoted by D n (t).
The queueing model is then, governed by the following set of differential-difference equations: The above equations are to be solved subject to the following boundary conditions: We assume that initially there are no customers in the system and the server is idle. So the initial conditions are

THE TIME-DEPENDENT SOLUTION
By using the supplementary variable technique, we obtain the following transient solution are the Laplace -Stieltjes transform of Service time B i (x) , vacation time V j (x), extended vacation W j (x), delay time H(x), and repair time U 1 (x) and U 2 (x) respectively.

THE STEADY STATE RESULTS
By using the property Let W q (z) be the PGF of queue size irrespective of the state of the system. Then we have, In order to obtain Q, using the normalization condition We see that for z=1, Wq(z) is indeterminate of the form 0 / 0. Therefore, we apply L'Hopital's rule and on simplifying we obtain the result (1), where C(1) = 1, C′(1) = E(I), is mean batch size of the arriving customers, ), is the mean of SOR time and let

THE MEAN QUEUE SIZE AND THE SYSTEM SIZE
We get the performance measure in the closed form.

PARTICULAR CASES
Our queueing system can be considered as a very general system and lot of systems already studied can be a particular case of our system.

Case (i) No Reneging
If we assume there is no reneging, single stage of service , single vacation policy and there is no second optional repair then The above result agrees with the result of Khalaf.R.F [5]

Case (ii) No Delay, No standby Server, No Extended Vacation, No Optional Repair
If we assume there is no delay, standby server, extended vacation and second optional repair with two stages of service having single vacation policy then The above result agrees with the result given Monita Baruah et al., [6]

NUMERICAL RESULT
In order to examine the validity of the results we approach numerical result. For that purpose we assume service time, vacation time, extended vacation time, delay time, FER and SOR are all follows exponential distribution and we choose the following arbitrary values : N =1 , M = 1 (single server and single vacation),

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This table clearly shows that as the delay time and the extended vacation time increases, the server idle time increases and the utilization factor decreases.

CONCLUSION
We have studied M [x] /G/1 queue with 'N' stages of services under different vacation policy and extended vacation subject to system breakdown with delay time and two phase of repairs. In addition we assume that the customers may renege during breakdown or vacation period due to impatience. The service time, vacation time, extended vacation time, delay time and repair are all follow general distribution. This work presents the closed form of the important performance measure. Many queueing system studied already are the special cases of this model.