The cyclic queue and the tandem queue

Abstract

We consider a closed queueing network, consisting of two FCFS single server queues in series: a queue with general service times and a queue with exponential service times. A fixed number N of customers cycles through this network. We determine the joint sojourn time distribution of a tagged customer in, first, the general queue and, then, the exponential queue. Subsequently, we indicate how the approach towards this closed system also allows us to study the joint sojourn time distribution of a tagged customer in the equivalent open two-queue system, consisting of FCFS single server queues with general and exponential service times, respectively, in the case that the input process to the first queue is a Poisson process.

Modeling Operational Flow Capacity and Evaluating Disaster Interventions for Fuel Distribution

“…Some disaster research explores scheduling of multicycle fleet and workers’ movements in steady state conditions in air force operated and charity run storehouses (Mattila et al., 2008; Urrea et al., 2019). They use simulation methods, since existing queueing theory literature does not have frameworks for multiserver tandem cyclic queues with time-limited customers, only single-server tandem cyclic queues (e.g., Campbell, 1991; Boxma and Daduna, 2014) or multiserver tandem queues (e.g., Kock et al., 2008; Baumann and Sandmann, 2017). Following the call for more humanitarian study validated with simulation (Kovacs and Moshtari, 2019), we extend this body of literature and apply queueing theory for steady state approximation and discrete event simulation for nonsteady state OFC measurement of multiserver tandem cyclic systems with time-limited customers.…”

State-Dependent Estimation of Delay Distributions in Fork-Join Networks

“…There is ample literature on delay estimations in service systems. An extensive survey of (mostly) steady-state results of exact delay expressions in queueing systems is given by Boxma and Daduna (1990). Another comprehensive survey, which adds coverage of real-time (state-dependent) results and the influence of delay information on customer behavior and system operations, is provided by Ibrahim (2018).…”

“…Stadje (1996) derived a closed-form solution of the Carmeli, Yom-Tov, and Boxma: Delay Distribution Estimation in Fork-Join Networks total sojourn time within a closed network of two singleserver queues in tandem with exponential service durations, given the number of customers in each queue at the time of arrival. Boxma (1983), Daduna (1986), and Boxma and Daduna (2014) yielded expressions for the joint LST of the sojourn time in the two tandem queues. Boxma (1983) considered the case of one queue with exponential service times followed by a queue with general service times in a closed network.…”

“…It was shown that reversing the order of the queues yields a different expression for the joint LST. Daduna (1986) generalized the service time distribution to an Erlang mixture, and Boxma and Daduna (2014) derived an expression for the joint LST of the sojourn times in both queues, using first general service times and then exponential service times (for both the closed and open network). We follow that approach and extend it to FJ queues, adding concurrency, which is prevalent in many service systems (e.g., healthcare systems), and thus providing an important additional layer to the analysis.…”

“…This again results in a model of two queues in tandem-one single-server queue (with either exponential or general service time distribution) followed by a single-or multiple-server queue with exponentially distributed service times. This open queueing model is a variation of the closed queueing model presented in Boxma and Daduna (2014). That is, we recursively calculate the LST by conditioning on the number of service completions at the second station during the remaining service time of the customer in service at the first station.…”