Journal of Statistical Theory and Practice

, Volume 4, Issue 1, pp 137–153 | Cite as

Batch Arrival Vacation Queue with Second Optional Service and Random System Breakdowns

  • Farzana A. MaraghiEmail author
  • Kailash C. Madan
  • Ken Darby-Dowman


A single server queue with a second optional service, Bernoulli schedule server vacations, and random system breakdowns was analyzed. It is assumed that customers arrive to the system in batches of variable size, but served one by one. As soon as the first service of a customer is completed, then with probability k he may opt for the second service. After a customer is served, the server may decide to leave the system with probability p, or to continue serving customers with probability 1 − p. If the system breaks down, it enters a repair process immediately. The repair time and the vacation time both are assumed to have general distributions. We obtain steady state results in explicit and closed form in terms of the probability generating functions for the number of customers in the queue, the average number of customers, and the average waiting time in the queue. Some special cases of interest are presented and a numerical illustration is provided.

AMS Subject Classification



Mx/G/1 queue Batch arrival Vacation time Bernoulli schedule Second optional service Random breakdown Steady state Queue size 


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Copyright information

© Grace Scientific Publishing 2010

Authors and Affiliations

  • Farzana A. Maraghi
    • 1
    Email author
  • Kailash C. Madan
    • 2
  • Ken Darby-Dowman
    • 1
  1. 1.School of Information Systems, Computing and MathematicsBrunel UniversityUxbridgeUK
  2. 2.College of Mathematical Sciences and ITAhlia UniversityManamaKingdom of Bahrain

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