Controlling the workload of M/G/1 queues via the q-policy

Abstract

We consider a single-server queueing system with Poisson arrivals and generally distributed service times. To systematically control the workload of the queue, we define for each busy period an associated timer process, {R(t), t ≥ 0}, where R(t) represents the time remaining before the system is closed to potential arrivals. The process {R(t), t ≥ 0} is similar to the well-known workload process, in that it decreases at unit rate and consists of up-jumps at the arrival instants of admitted customers. However, if X represents the service requirement of an admitted customer, then the magnitude of the up-jump for the timer process occurring at the arrival instant of this customer is (1 − q)X for a fixed q ∈ [0, 1]. Consequently, there will be an instant in time within the busy period when the timer process hits level zero, at which point the system immediately closes and will remain closed until the end of the current busy period. We refer to this particular blocking policy as the q-policy. In this paper, we employ a level crossing analysis to derive the Laplace-Stieltjes transform (LST) of the steady-state waiting time distribution of serviceable customers. We conclude the paper with a numerical example which shows that controlling arrivals in this fashion can be beneficial.

Waiting Time Distributions in the Preemptive Accumulating Priority Queue

“…Our method of analysis used the maximal priority process of the PAPQ and related it to the maximal priority process of the FCFS M/G/1 queue with accumulating priority and blocking, as introduced by Fajardo and Drekic (2015). We stress that this approach mimics that used for the analysis of static preemptive priority models where one relates the virtual wait process in those models to the virtual wait process of the classical FCFS M/G/1 queue (e.g., see Brill 2008).…”

“…In order to show this, we make an important connection between (M k (t), M k+1 (t)) during level-k accreditation intervals and the maximal priority process of the M/G/1 queue with accumulating priority and blocking introduced by Fajardo and Drekic (2015). This model represents a FCFS M/G/1 queue, whose customers, upon arrival to the system, accumulate priority linearly at rate ξ 1 > 0.…”

A reliability-based maintenance technicians’ workloads optimisation model with stochastic consideration

“…This is necessary avert machine breakdowns, which could be more costly. Fajardo and Drekic (2015) pointed out that cost-effective maintenance checks could be achieved during close-down periods. The problem of technician cost and equipment availability was studied by Safaei et al (2008).…”

On a general mixed priority queue with server discretion

“…(20) was first presented and proven by Stanford et al [20]. Fajardo and Drekic [12] later provided an alternate derivation of the result using level-crossing methodology for a related M/G/1 queue working under a particular blocking policy known as the q-policy. To explain Eq.…”