Queueing Mathematics 1

(updated  17th March 2008)

Keywords: simulating people, simulating crowds, simulating crowd dynamics

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The Single-Channel Queueing model with Poisson arrivals and exponential service times.

There are many books written about the mathematics of queueing theory. The basic principles are that queues have an arrival rate, a service rate, and a discipline. The accepted method of defining a queue uses the following symbols.

= Expected number of arrivals per time period (mean arrival rate)
= Expected number of services possible per time period (mean service rate)

Using the following assumptions:-

  • The queue has a single channel
  • The pattern of arrivals follow a Poisson probability distribution
  • The service times follow an exponential probability distribution
  • The queue discipline is first come, first served
Probability that there is no-one in the queue
Probability of N people in the system
The average number of people in the queue
The average number of people in the system
The average time a person spends waiting in the queue
The average time a person spends in the system
The probability that a person has to wait in the queue.

All of the above calculations can be integrated into a spreadsheet model to provide the user with the critical information about queueing duration, size, length of wait. etc.