Infinite Server Queues with Synchronized Departures Driven by a Single Point Process

Abstract

We analyze an infinite-server queueing model with synchronized arrivals and departures driven by the point process {T n } according to the following rules. At time T n , a single customer (or a batch of size β n ) arrives to the system. The service requirement of the ith customer in the nth batch is σ i,n . All customers enter service immediately upon arrival but each customer leaves the system at the first epoch of the point process {T n } which occurs after his service requirement has been satisfied. For this system the queue length process and the statistics of the departing batches of customers are investigated under various assumptions for the statistics of the point process {T n }, the incoming batch sequence {β n }, and the service sequence {σ i,n }. Results for the asymptotic distribution of the departing batches when the service times are long compared to the interarrival times are also derived. Consider a system where groups (or batches) of customers arrive at the epochs of a point process {T n ; n ∈ Z} defined on the whole real line. The nth group arrives at time T n and consists of β n customers. The ith customer of the nth group, which we will denote by C i,n , 1 ≤ i ≤ β n , n ∈ Z, remains in the system for σ i,n time units, and then departs at the next arrival point after his serviceIn more descriptive terms we envision a shuttle bus which arrives at a certain facility at the epochs {T n } of a stationary and ergodic point process. Each time the shuttle bus arrives, it brings along a new group of passengers and delivers them to the facility. The ith passenger of the nth group will stay in this facility for σ i,n time units, and then he will move on to a waiting area from where he will be picked up by the first shuttle that arrives. We assume that the facility, the waiting area, and the shuttle, all have infinite capacity so that a new group of passengers can always be delivered to the 1

Tollbooth tandem queues with infinite homogeneous servers

“…CHAO ET AL. for example, Zazanis (2004), studied an infinite server queue where departures are synchronized by a given point process; in the tollbooth tandem queue, the departure of a customer is affected by departures of the customers ahead of himlher which depends on the state of the system. The tollbooth tandem queue is also different from the asymmetric simple inclusion process (ASIP) studied by, for example, Reuveni et al (2012Reuveni et al ( ), (2014, in which each server opens its gate to accept customers from an up-stage server according to a Poisson process.…”

Tollbooth tandem queues with infinite homogeneous servers

“…The tollbooth tandem queue also differs from some of the studies on MlG/oo queues with synchronized departures, 942 x. CHAO ET AL. for example, Zazanis (2004), studied an infinite server queue where departures are synchronized by a given point process; in the tollbooth tandem queue, the departure of a customer is affected by departures of the customers ahead of himlher which depends on the state of the system. The tollbooth tandem queue is also different from the asymmetric simple inclusion process (ASIP) studied by, for example, Reuveni et al (2012), (2014), in which each server opens its gate to accept customers from an up-stage server according to a Poisson process.…”