Queueing Models and Simulation Techniques

Queueing Models

M/M/s: Exponential arrival and service with more than one server. These models are needed in the economic analysis of the system.

M/G/1: Exponential arrival with general service time.

M/D/1: Exponential arrival with deterministic (fixed) service time. Because of lower variability in service time, performance measures are better than M/M/1.

Queues with priority: Performance measures will improve for high-priority customers and worsen for low-priority customers. However, performance measures averaged across all customers (high AND low priority) will remain the same as before.

Distinguish between the number of servers and the number of queues:

1. An M/M/2 queue with two servers but a single queue of waiting customers.
2. Two M/M/1 queues where there are two servers, but each server has its own queue of waiting customers. (Change λ)

Simulation Techniques

Uniform distribution: With values between a and b equally likely: a + (b – a)*RAND()

Exponential distribution: – (mean) * LN(RAND())

Normal distribution: With mean

and standard deviation : NORMINV(RAND(), , ).

Outcomes of simulation modeling are statistical measures such as averages. Statistical results are typically subjected to additional statistical analysis to determine their degree of accuracy.

Confidence Limits

Confidence limits are developed for the analysis of the statistical validity of simulation results:

Formulas for 95% confidence limits:

• Upper confidence limit: + 1.96
• Lower confidence limit: - 1.96

is the mean, and s is the standard deviation from a sample of size n from any population. We can be 95% confident that the true population mean will be between the upper confidence limit and the lower confidence limit.

Simulation Procedure

1. Formulate problem.
2. Specify probabilistic input variables and assign probability distributions to input variables.
3. Implement model: (Generating values for random variables from probability distribution; Compute output variables and record results)
4. Repeat step 3 until a sufficient number of trials to generate a useful distribution of outputs.