COMPUTATIONAL APPROACH FOR TRANSIENT BEHAVIOUR OF M/M (a, b)/1 BULK SERVICE QUEUEING SYSTEM WITH WORKING VACATION

A new computational technique is used to evaluate the Transient behaviour of Single Server Bulk Service Queueing System with Working Vacation with arrival rate λ which follows a Poisson process and the service will be in bulk. In this model the server provides two types of services namely normal service and lower service. The normal service time follows an exponential distribution with parameter μ1. The lower service rate follows an exponential distribution with parameter μ2. The vacation time follows an exponential distribution with parameter α. According to Neuts, the server begins service only when a minimum of ‘a’ customers in the waiting room and a maximum service capacity is ‘b’. An infinitesimal generator matrix is formed for all transitions. Time dependent solutions and Steady state solutions are acquired by using Cayley Hamilton theorem. Numerical studies have been done for Time dependent average number of customers in the queue, Transient probabilities of server in vacation and server busy for several values of t, λ, μ1, μ2, α, a and b.


INTRODUCTION
The main objective of this research paper is to analyze the transient behaviour of Bulk service queueing system with working vacation by computational approach. Bulk Queues defined as queues with batch arrivals or batch services or both in batches. The batch size may be fixed or a random variable. In this model, arrival is single but service is in batch. Bulk service queues have potential applications in many areas e.g. in traffic signal systems, in restaurants, cinema halls, in transportation processes involving buses, airplanes and so on. Bernoulli vacation with reneging of customer (2020).

THE MATHEMATICAL MODEL AND ITS SOLUTIONS
A new computational method is used to estimate the Transient behaviour of Single server Bulk service queueing system with working vacation whose arrival rate λ follows a Poisson process and the service will be in bulk with normal service time follows an exponential distribution with parameter μ1 and the lower service rate in working vacation follows an exponential distribution with parameter μ2. The vacation time follows an exponential distribution with parameter α.
Assuming that there are 'a' customers in the system at time t = 0. The general considerations for bulk service queueing system with working vacation are • After completion of the normal service if the number of customers in the queue is less than 'a' then the server goes for working vacations and becomes idle and start the service with lower service rate only if the batch size reaches 'a'. If the number of customers in the queue lies between 'a' and 'b' then all the customers in the queue will be taken for service and queue becomes empty and server starts normal service. If there are more than 'b' customers are waiting in the queue then the first 'b' customers are taken for service and the remaining customers will have to wait for service.
• During the working vacation period, if he finds the number of customers in the queue is between 'a' and 'b' then all the customers in the queue will be taken for service with lower service rate and queue becomes empty. If he finds the number of customers in the queue is more than b then the first b customers will be taken from the queue for service with lower service rate and remaining customers will be in the queue.
• After completion of the vacation period the server comes back to the system and starts normal service only if there are a minimum of 'a' customers in the queue, if the server finds less than 'a' customers in the queue then he will be waiting in the system for normal service. If he finds the number of customers in the queue is between 'a' and 'b' then all the customers in the queue will be taken for normal service and queue becomes empty and server starts normal service. If he finds more than 'b' customers are waiting in the 2560 S. SHANTHI, A. MUTHU GANAPATHI SUBRAMANIAN, GOPAL SEKAR queue then the first 'b' customers are taken for normal service and the remaining customers will have to wait for normal service.
Remaining all other entries are zero.
Further, we can write the above equations (1), (2) and (3) as where Where Solving the above set of equation we get, When t=0,

DESCRIPTION OF COMPUTATIONAL METHOD
The following effective computational procedure is used to find the Time dependent probabilities of number of customers in the queue at time t. The time dependent probabilities vector is denoted by Step 1: Assume that the matrix Q is finite that is the number of customers in the queue at time t is M (sufficiently large). The value of M can be chosen so that the loss probability is small. Due to the intrinsic nature of the system, the only choice available for studying M is through X t P P P P P P P P P P P P P P P P X t P P P P P P P P P P P P P P P P S. SHANTHI, A. MUTHU GANAPATHI SUBRAMANIAN, GOPAL SEKAR algorithmic methods. While a number of approaches are available for determining the cut-off point, M, the one that seems to perform well is to increase M until the largest individual change in the elements of X (t) for successive values is less than ε a predetermined infinitesimal value.
Step 2: Find the Eigen values of this finite order matrix tQ T .
Step 4: Use these Eigen values in the Vandermonde's matrix .
Step 5: Let Step 6: Find and we get .
Step 7: Using in Step 8: Extract the first column of this Exponential matrix tQ T and store in X (t).
Step 9: This probability vector X (t) provides time dependent probabilities of number of customers in the queue at time t.

SYSTEM PERFORMANCE MEASURES
The following system measures are used to bring out the Transient behaviour of bulk service queueing model with working vacation under study. Numerical study has been dealt in very large scale to study the following measures for several values of t, λ, µ1, µ2, α, a and b.
a. Probability that there are n customers in the queue when the server is idle at time t = . . . and . . .  Table 1 to Table 4 show Transient probabilities of number of customers in the queue when the server is idle for several values of t, λ, µ1, µ2, α, a and b. We infer the following • The sequence Table 5 to Table 8 show Transient probabilities of number of customers in the queue when the server is busy for several values of t, λ, µ1, µ2, α, a and b. We infer the following • As the value of t increases the Transient Probabilities • The sequence Table 9 to Table 12 show Transient probabilities of number of customers in the queue when the server is idle in working vacation period for several values of t, λ, µ1, µ2, α, a and b. We infer the following

• As the value of t increases the Transient Probabilities
• The sequence Table 13 to Table 16 show Transient probabilities of number of customers in the queue when the server is busy in working vacation period for several values of t, λ, µ1, µ2, α, a and b. We infer the following

• As the value of t increases the Transient Probabilities
• The sequence