Queueing - Inevitable fact of live
(updated 31st August 2007)
Keywords: simulating people, simulating crowds, simulating crowd dynamics
There are two main features of queue: arrival rate and service rate.
The arrival rate is the frequency and distribution of the things arriving for a service. This could be frequency (how often) and distribution (how many arrive at once or across time) of busses arriving to pick people up at a bus stop (where the time at the bus stop is the service rate). Or it could mean the number of people arriving per hour (average) to be served at a bank and the time (average) it takes to serve each person. Both of these are examples of queueing systems and the mathematics are not obvious as the above article clearly indicates. That is why the Great British tradition of queueing has evolved - the people who count the cost of an event think in linear terms (input = output) but as you can see above - queue's don't behave in a linear manner.
The spreadsheet model below shows the features of a singe channel, first come first serve, Poisson arrival rate and exponential service rate queue. Whoa.....did I just sneak a complex mathematical description in the text....Guilty as accused. Let me try and simplify that last sentence. A Poisson distribution is sometimes called a "normal" distribution in that most things arrive at an average rate (say 1 per minute) but some things arrive very quickly (say 1 per second) and some arrive very slowly say 1 per 10 minutes).
So a Poisson distribution has a curve, sometimes called a bell shaped curve and looks something like this.:
Poisson Probability Distribution
The exponential service rate means that most people do things quickly, but some take absolute ages (how often have you observed that at a supermarket checkout). The exponential curve looks something like this:
Exponential Curve
Taking a variety of curves (either side of the anticipated arrival rate) allows the model builder to create an "envelope" of anticipated arrival profiles.
This can be easily constructed in a spreadsheet model. Crowd Dynamics Limited run workshops on how to construct and use these types of models (email us for further information)
The spreadsheet above indicates the critical factors in a queue. The mathematics are all published in a number of highly technical books and journals but applying them to specific locations can be complex. If you have any questions about the application of queueing theory - drop me a line - I'll be happy to answer questions.
Our arrival rate can be estimated (average) and our service rate can also be estimated (again an average). So we would expect that if 100 people arrived per hour and we could service 100 people per hour then things would be just simply perfect. Right ?
WRONG !
If we examine the depth of the queue using the spreadsheet morel we can see the problem. As the arrival rate increases (in this example by an average of 1 person every 30 seconds) we can see that there is a massive increase in the queue depth. We can serve 10 people per minute but when 9 people per minute arrive we have a queue of average depth of 7-8 people ! The graph below illustrates the problems with queues - they get very large very quickly.
If you balance inputs to outputs then the first time something takes a little longer than average - you have a queue. Furthermore that queue will not go away until the arrival rate DROPS below average or a series of services take less than average to perform. That is why there are always queues - because queueing systems are often assumed to be linear. Accountants like to balance inputs with outputs (they hate to see someone idle - it's a waste of money). However, the reason to have an idle server is to cope with the above average arrivals. In other words - you NEED some slack in the system to ensure that queues are manageable.
In practice people get queues more wrong that right. The mathematics is difficult but some simple models can help understand the dynamics of queues. The author has many years experience with modelling queueing system - from production and operations management where nearly every process involved has a queueing problem.