Extremal properties of the FIFO discipline in queueing networks

Abstract

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.

Constant time per edge is optimal on rooted tree networks

“…This approach, however, has certain limitations: it requires Poisson arrivals into the network, the comparison holds only between constant and exponential service times, and the bounds apply only to expected delay. Results similar to but more general than those of [20] and [4] were also discovered independently by Righter and Shanthikumar in [14] using more general stochastic comparisons.…”

“…These techniques have also been applied and generalized to more complicated processes [1,13,14,19,22]. Indeed, although our work was derived independently, it strongly resembles the work of Niu [13], who examined tandem queues using a similar approach.…”

On the Gittins index in the M/G/1 queue

“…However, for IHR distributions, FCFS does not minimize the number in the stochastic sample path sense. For this, ILR (Increasing in Likelihood Ratio) distributions are required [14]. Also, for the DHR, IHR, ILR and NBUE results, the arrival process can be arbitrary.…”

Ordering New Conditional Residual Lifetimes ofk-out-of-nSystems

“…For more details on DLR and ILR, see Shaked and Shanthikumar (1987) and Righter and Shankthikumar (1992). Other properties of DFR can be found in Barlow and Proschan (1981).…”