Queue modeling probability an arrival would not have to wait for service
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Queuing theory
Queuing theory deals with problems which involve queuing (or waiting). Typical examples might be:
banks/supermarkets - waiting for service
computers - waiting for a response
failure situations - waiting for a failure to occur e.g. in a piece of machinery
public transport - waiting for a train or a bus
As we know queues are a common every-day experience. Queues form because resources are limited. In fact it makes economic sense to have queues. For example how many supermarket tills you would need to avoid queuing? How many buses or trains would be needed if queues were to be avoided/eliminated?
In designing queueing systems we need to aim for a balance between service to customers (short queues implying many servers) and economic considerations (not too many servers).
In essence all queuing systems can be broken down into individual sub-systems consisting of entities queuing for some activity (as shown below).
Typically we can talk of this individual sub-system as dealing with customers queuing for service. To analyse this sub-system we need information relating to:
arrival process:
how customers arrive e.g. singly or in groups (batch or bulk arrivals)
how the arrivals are distributed in time (e.g. what is the probability distribution of time between successive arrivals (the interarrival time distribution))
whether there is a finite population of customers or (effectively) an infinite number
The simplest arrival process is one where we have completely regular arrivals (i.e. the same constant time interval between successive arrivals). A Poisson stream of arrivals corresponds to arrivals at random. In a Poisson stream successive customers arrive after intervals which independently are exponentially distributed. The Poisson stream is important as it is a convenient mathematical model of many real life queuing systems and is described by a single parameter - the average arrival rate. Other important arrival processes are scheduled arrivals; batch arrivals; and time dependent arrival rates (i.e. the arrival rate varies according to the time of day).