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Operations Research Models and Methods
Models Section
Model Classes
Non-Markov Systems

Analytical formulas are available for a few situations that do not require the Markov assumption for either the input or service process. These are useful because in practical instances, interarrival and service time distributions may be not be reasonably approximated by exponential distributions. For the M/G/1 system the formulas give accurate results. For all other systems the formulas are approximate. There are no formulas available for finite queue or finite population systems.

Non-Markov systems require the specification of the coefficients of variation (COV) of the arrival and service processes. The COV for arrivals is the standard deviation for time between arrivals divided by the mean time between arrivals . For an exponential distribution the COV is 1. Distributions with less variability than the exponential have COV < 1, while distributions with more variability have COV > 1. The service process COV is the standard deviation of the service time divided by the mean service time.



Example: An order picking process

An order picking process in a warehouse gets calls for service at an average rate of 8.5 per hour. The average time to fill the order is 0.1 hours. For analysis purposes assume both times are exponentially distributed. Analyzing the system as an M/M/1 queue, the average time in the queue is 0.5667 hours. An opportunity arises to reduce the variability of the process for filling orders. The inventory manager wonders if the change is worth the cost.


The set of results for the Non-Markovian case is smaller than those available for Poisson queues. This is partially due to the restriction against finite queues and finite populations, making some results not relevant. The approximations do not allow the computation of state probabilities.

The results for three replications of the queuing model are shown below. The display was created by the Non-Markov option of the Queuing add-in. To illustrate the effect of reducing the variability in the service times, we set the COV of the service process to 0.5 in the second model and 0 in the third (no variability). In general the numbers and time in the queue decrease as variability decreases. Note that with COV of 1, the distribution is actually Markovian (Poisson process). Thus we see an M in the type designation when the COV is 1. The G in the type of the second and third models indicates that the service process has a general distribution.

It is clear from the results that reducing variability in the service process causes decreased time in the queue. The mean time in service does not change because that is fixed by the data and unaffected by the variability. The efficiency only depends on the amount of work available compared to the number of servers, so it does not change with variability.


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Operations Research Models and Methods
by Paul A. Jensen
Copyright 2004 - All rights reserved