The Single Server Queue with Synchronized Services

• Published in: Stochastic models Vol. 26; no. 4; pp. 617 - 648
• Main Authors: Economou, Antonis, Kapodistria, Stella, Resing, Jacques
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Taylor & Francis Group 05.11.2010; Taylor & Francis; Taylor & Francis Ltd
• Subjects: Busy period; Conditional residual service time; Distribution theory; Embedded distribution; Equilibrium distribution; Exact sciences and technology; Markov analysis; Mathematics; Primary 60K25; Probability and statistics; Probability theory and stochastic processes; Queueing system; Queuing theory; Repeated services; Sciences and techniques of general use; Secondary 90B22; Sojourn time; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Stochastic analysis; Stochastic models; Stochastic processes; Studies; Synchronized services; Stochastic model; Conditional distribution; Repeated services; Conditional residual service time; Probability theory; Embedded distribution; Primary 60K25; Probability; Sojourn time; Continuous time; Probability distribution; Distributed system; Synchronization; Single server queue; Poisson process; Sojourn time distribution; Service time; Queueing system; Equilibrium distribution; Busy period; Distribution function; Secondary 90B22; Synchronized services
• ISSN: 1532-6349; 1532-4214
• DOI: 10.1080/15326349.2010.519670

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.