Application to Single Server Queues

The attained waiting time of customers in service of the G/G/1 queue is compared for various work-conserving service disciplines. It is proved that the attained waiting time distribution is minimized (maximized) in convex order when the discipline is FCFS (PR-LCFS). We apply the result to characterize finiteness of moments of the attained waiting time in the GI/GI/1 queue with an arbitrary work-conserving service discipline. In this discussion, some interesting relationships are obtained for a PR-LCFS queue.

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Mean Buffer Contents in Discrete-Time Single-Server Queues with Heterogeneous Sources

Queueing Systems ◽

10.1007/s11134-005-6480-0 ◽

2005 ◽

Vol 49 (2) ◽

pp. 161-186 ◽

Cited By ~ 1

Author(s):

Tetsuya Takine

Keyword(s):

Discrete Time ◽

Single Server ◽

Single Server Queues

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Extremal properties of the FIFO discipline in queueing networks

Journal of Applied Probability ◽

10.2307/3214728 ◽

1992 ◽

Vol 29 (4) ◽

pp. 967-978 ◽

Cited By ~ 14

Author(s):

Rhonda Righter ◽

J. George Shanthikumar

Keyword(s):

Likelihood Ratio ◽

Service Time ◽

Queueing Networks ◽

Service Rate ◽

Service Discipline ◽

Single Server ◽

Single Server Queues ◽

First Results ◽

Service Times ◽

Number Of Customers

We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons:(i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means.(ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means.We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR.Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.

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