M/M/1 Queue Model
Introduction
A waiting line or queue naturally forms when a demand exceeds the resources that can serve that demand. Whether this is bank tellers or grocery clerks serving numerous customers, or parts arriving in a factory waiting for an operation to be performed, there are many occurrences of queues in everyday like.
Queueing theory is the study of waiting lines from a mathematical perspective. With a defined process flow and known arrival and service rates for entities, the queue lengths and waiting times can be predicted. We can also answer questions such as what are the probabilities of n entities in the system, and what is the utilization of the servers.
When I began working line balancing and designing workcells layouts in a factory environment, it became important to understand how queues formed and how we might predict how they would impact operations.
Simple Queueing Model
In queueing theory, there are many different models to represent different levels of scenarios and complexity. The M/M/1 model is the most basic. In this model entities arrive into the system, are processed by a single server, and then exit. The arrival and service rates follow an exponential distribution. Another similar model which allows for multiple servers is the M/M/c model.
M/M/1 Queue Example:
Parts come to a processing station at a mean inter-arrival rate of one every 30 seconds (0.5 minutes) and have a mean processing time of 15s (0.25 minutes). In this case we’d want to convert the inter-arrival time to arrival rate by taking the reciprocal value. Thus λ = 1 / 0.5 = 2 and μ = 1 / 0.25 = 4. Substituting these values into the M/M/1 queuing model yields the results