The Single Server Queue with Synchronized Services

Abstract

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.

Keywords:

• Busy period
• Conditional residual service time
• Embedded distribution
• Equilibrium distribution
• Queueing system
• Repeated services
• Sojourn time
• Synchronized services

Mathematics Subject Classification:

• Primary 60K25
• Secondary 90B22

Notes

(λ = 1, E[B] = 1).