We consider a single server Markovian queue with setup times. Whenever this system becomes empty, the server is turned off. Whenever a customer arrives to an empty system, the server begins an exponential setup time to start service again. We assume that arriving customers decide whether to enter the system or balk based on a natural reward-cost structure, which incorporates their desire for service as well as their unwillingness to wait. 

We examine customer behavior under various levels of information regarding the system state. Specifically, before making the decision, a customer may or may not know the state of the server and/or the number of present customers. We derive equilibrium strategies for the customers under the various levels of information and analyze the stationary behavior of the system under these strategies. We also illustrate further effects of the information level on the equilibrium behavior via numerical experiments.

Equilibrium customer strategies in a single server Markovian queue with setup times

This paper deals with the analysis of an M/M/c queueing system with setup times. This queueing model captures the major characteristics of phenomena occurring in production when the system consists in a set of machines monitored by a single operator. We carry out an extensive analysis of the system including limiting distribution of the system state, waiting time analysis, busy period and maximum queue length.

Analysis of a Multiserver Queue with Setup Times

Optimal balking strategies in single-server queues with general service and vacation times

A batch arrival queueing system with a single vacation between two successive busy periods and with exhaustive service is considered. The departure process h(t) is studied first on a single vacation cycle. The approach based on renewal theory is applied to obtain results in the general case. In particular, the explicit representation for the generating function of Laplace transform of the probability function of h(t) is derived. All formulae are written in terms of input parameters of the system and factors of a certain canonical factorization of Wiener–Hopf type. A numerical approach to results is discussed as well.

Some New Results for Departure Process in the M X /G/1 Queueing System with a Single Vacation and Exhaustive Service

In the article the queueing system of GI/G/1 type with batch arrival of customers and a single exponentially distributed vacation period at the end of every busy period is considered. Basic characteristics of transient state of the system are investigated: the first busy period, the first vacation period and the number of customers served during the first busy period. New results for the Laplace transform of the joint distribution of these three variables are obtained in dependence on the initial conditions of the system.

GI/G/1/∞ batch arrival queueing system with a single exponential vacation

This paper deals with the steady state behaviour of an M⁄M⁄c queueing system with a general startup time under N-policy, where the system is turned off as soon as the system becomes empty and turned on again when the queue size becomes ‘N’. The service of the first customer in each busy period is always preceded by a random startup time (during which no proper work is done) to gearup the system. We place our emphasis on understanding the operational characteristic of the queueing system and derive the probability generating functions of the queue size distribution and joint distribution of the number of arrivals and startup time. Also, we provide appropriate interpretation of the queue size distribution and other performance measures. Finally, we verify some results of M⁄M⁄1 queueing system with general startup time under N-policy.

http://csa.sagepub.com/content/49/3-4/199.full.pdf

A Multiserver Poisson Queue with a General Startup Time Under N-Policy:

We consider a single server Markovian queueing system with compound Poisson arrivals of batches of random size under a single vacation policy, where the server takes exactly one vacation between two successive busy periods. We are concerned with the steady state distributions of the additional queue size, additional delay and queue waiting time distribution.

Gautam Choudhury

Papers

A Multiserver Poisson Queue with a General Startup Time Under N-Policy:

Analysis of a Batch Arrival Poisson Queue Under Single Vacation Policy