The Single Server Queue with Synchronized Services

We consider a single server queueing system with generally distributed synchronized services. More specifically, customers arrive according to a Poisson process and there is a single server that provides service, if there is at least one customer present in the system. Upon the initialization of a service, all present customers start to receive service simultaneously. We consider the gated version of the model, that is, customers who arrive during a service time do not receive service immediately but wait for the beginning of the next service time. At service completion epochs, all served customers decide simultaneously and independently whether they will leave the system or stay for another service. The probability that a served customer gets another service is the same for all customers. We study the model and derive its main performance measures that include the equilibrium distribution of the number of customers at service completion epochs and in continuous time, the busy period and the sojourn time distributions. Moreover, we prove some limiting results regarding the behavior of the system in the extreme cases of the synchronization level. Several variants and extensions of the model are also discussed.

[1]  J. Gani,et al.  Birth, immigration and catastrophe processes , 1982, Advances in Applied Probability.

[2]  Antonis Economou,et al.  Synchronized abandonments in a single server unreliable queue , 2010, Eur. J. Oper. Res..

[3]  A. G. Pakes,et al.  Some Conditions for Ergodicity and Recurrence of Markov Chains , 1969, Oper. Res..

[4]  J. Kingman.  A FIRST COURSE IN STOCHASTIC PROCESSES , 1967 .

[5]  Yoav Kerner.  The Conditional Distribution of the Residual Service Time in the M n /G/1 Queue , 2008 .

[6]  Antonis Economou,et al.  The compound Poisson immigration process subject to binomial catastrophes , 2004, Journal of Applied Probability.

[7]  Samuel Karlin,et al.  A First Course on Stochastic Processes , 1968 .

[8]  Eitan Altman,et al.  Analysis of customers’ impatience in queues with server vacations , 2006, Queueing Syst. Theory Appl..

[9]  Antonis Economou,et al.  Synchronized reneging in queueing systems with vacations , 2009, Queueing Syst. Theory Appl..

[10]  Marcel F. Neuts.  AN INTERESTING RANDOM WALK ON THE NON-NEGATIVE INTEGERS , 1994 .

[11]  Antonis Economou,et al.  Alternative Approaches for the Transient Analysis of Markov Chains with Catastrophes , 2008 .

[12]  Uri Yechiali,et al.  Queues with system disasters and impatient customers when system is down , 2007, Queueing Syst. Theory Appl..

[13]  J R Artalejo,et al.  Evaluating growth measures in an immigration process subject to binomial and geometric catastrophes. , 2007, Mathematical biosciences and engineering : MBE.

[14]  H. C. Tijms,et al.  A simple proof of the equivalence of the limiting distributions of the continuous-time and the embedded process of the queue size in the m/g/1 queue , 1976 .

[15]  S. Lang.  Complex Analysis , 1977 .

[16]  W. D. Ray,et al.  Stochastic Models: An Algorithmic Approach , 1995 .

[17]  Antonis Economou,et al.  q-SERIES IN MARKOV CHAINS WITH BINOMIAL TRANSITIONS , 2008, Probability in the Engineering and Informational Sciences.