Queueing model
In queueing theory, a queueing model is used to approximate a real queueing situation or system, so the queueing behaviour can be analysed mathematically. Queueing models allow a number of useful performance measures to be determined, including:
- the average number in the queue, or the system,
- the average time spent in the queue, or the system,
- the statistical distribution of those numbers or times,
- the probability the queue is full, or empty, and
- the probability of finding the system in a particular state.
These performance measures are important as issues or problems caused by queueing situations are often related to customer dissatisfaction with service or may be the root cause of economic losses in a business. Analysis of the relevant queueing models allows the cause of queueing issues to be identified and the impact of any changes that might be wanted to be assessed.
Notation
Queueing models can be represented using Kendall's notation:
- A/B/S/K/N/Disc
Where:
- A is the interarrival time distribution
- B is the service time distribution
- S is the number of servers
- K is the system capacity
- N is the calling population
- Disc is the service discipline assumed
Some standard value of the notation include:
- M for a Markovian (exponential) distribution
- Eκ for an Erlang distribution with κ phases
- D for Deterministic (constant)
- G for General distribution
Models
Single Server Queue
Single server queues are, perhaps, the most commonly encountered queueing situation in real life. One encounters a queue with a single server in many situations, including business (e.g. sales clerk), industry (e.g. a production line), transport (e.g. a bus, a taxi rank, an intersection), telecommunications (e.g. Telephone line), computing (e.g. processor sharing). Even where there are multiple servers handling the situation it is possible to consider each server individually as part of the larger system, in many cases. (e.g A supermarket checkout has several single server queues that the customer can select from.) Consequently, being able to model and analyse a single server queue's behaviour is a particularly useful thing to do.
Poission Arrivals and Service
M/M/1// represents a single server that has unlimited queue capacity and infinite calling population, both arrivals and service are Poisson (or random) processes, meaning the statistical distribution of both the inter-arrival times and the service times follow the exponential distribution. Because of the mathematical nature of the exponential distribution, a number of quite simple relationships are able to be derived for several performance measures based on knowing the arrival rate and service rate.
This is fortunate because, an M/M/1 queuing model can be used to approximate many queuing situations.
Poission Arrivals and General Service
M/G/1// represents a single server that has unlimited queue capacity and infinite calling population, while the arrival is still Poisson process, meaning the statistical distribution of the inter-arrival times still follow the exponential distribution, the distribution of the service time does not. The distribution of the service time may follow any general statistical distribution, not just exponential. Relationships are still able to be derived for a (limited) number of performance measures if one knows the arrival rate and the mean and variance of the service rate. However the derivations a generally more complex.
A number of special cases of M/G/1 provide specific solutions that give broad insights into the best model to choose for specific queueing situations because they permit the comparison of those solutions to the performance of an M/M/1 model.