This work aims at studying the fairness of multi-queue and multi-server queueing systems. We deal with the issues of queue-multiplicity, queue joining policy and queue jockeying and use a quantitative measure (RAQFM) to evaluate them. Our results yield the relative fairness of the mechanisms as a function of the system configuration and parameters. Practitioners can use these results to quantitatively account for system fairness and to weigh efficiency aspects versus fairness aspects in designing and controlling their queueing systems. In particular, we quantitatively demonstrate that: 1) Joining the shortest queue increases fairness, 2) A single "combined" queue system is more fair than "separate" (multi) queue system and 3) Jockeying from the head of a queue is more fair than jockeying from its tail.

Fairness is a major issue in the operation of queues, perhaps it is the reason why queues were formed in the first place. Recent studies show that the fairness of a queueing system is important to customers not less than the actual delay they experience. Despite this observation little research has been conducted to study fairness in queues, and no commonly agreed upon measure of queue fairness exists. Two recent research exceptions are Avi-Itzhak and Levy [1], where a fairness measure is proposed, and Wierman and Harchol-Balter [18] (this conference, 2003), where a criterion is proposed for classifying service policies as fair or unfair; the criterion focuses on customer service requirement and deals with fairness with respect to service times.In this work we recognize that the inherent behavior of a queueing system is governed by two major factors: Job seniority (arrival times) and job service requirement (service time). Thus, it is desired that a queueing fairness measure would account for both. To this end we propose a Resource Allocation Queueing Fairness Measure, (RAQFM), that accounts for both relative job seniority and relative service time. The measure allows accounting for individual job discrimination as well as system unfairness. The system measure forms a full scale that can be used to evaluate the level of unfairness under various queueing disciplines. We present several basic properties of the measure. We derive the individual measure as well as the system measure for an M/M/1 queue under five fundamental service policies: Processor Sharing (PS), First Come First Served (FCFS), Non-Preemptive Last Come First Served (NP-LCFS), Preemptive Last Come First Served (P-LCFS), and Random Order of Service (ROS). The results of RAQFM are then compared to those of Wierman and Harchol-Balter [18], and the quite intriguing observed differences are discussed.

A time-sharing queue serving a finite number of customers is described. It is assumed that both the service time and the time elapsing between termination of service and the next arrival of the same customer at the queue (service station) are exponential. The model was studied by Krishnamoorthi and Wood, but their results are not in complete agreement with the results of this paper. In addition, some new results are presented in terms of steady-state expectations.