The queuing theory speaking on a general note is a mathematical theory of waiting lines, a model of which helps in calculating approximate lengths of queues and waiting period. Having usage in multiple areas such as computing of data, telecommunication and traffic engineering, this is one theory that has garnered immense popularity in recent times and is waiting for further exploitation in future.
However, to understand details associated with this term, it is very important to understand, specific terms that are associated with this theory as a whole. This clarity would result in better utilisation of this theory.
Specific terms associated with this theory:
- The rate of arrival:
This is the pattern in which consumers come to the specific service centres to get their doubts clarified. There are generally two patterns, haphazard and regular, wherein regular pattern of arrival by customers is quite less in comparison to haphazard arrival of customers for servicing.
- Poisson’s Distribution:
Specifically used in probability and statistical analysis, is a form of discrete probability distribution which depicts the probability of a given number of events within a specific set of time. In this case, it is associated with the probable arrival of customers during a specific time period, and the remainder pattern of arrival of customers specifically follows this format.
In this mathematical calculation, variance is equal to mean and it is the Greek letter λ (lambda) that is used for denoting it.
Notation used: P(n) = Probability of n arrivals (customers)λ = Mean arrival rate = Constant. This is known as factorial and follows the usual routine.
Though as a matter of fact, Poisson’s Distribution can be used in multiple areas and mathematical calculations, however, it has certain limitations. The primary limitation associated with this is its assumption that arrival occurs at a random rate and is totally independent of the variable aspects associated with this. Also, it is assumed that other parameters are not taken into consideration while dealing with this prospect.
However, such is not the case! Arrival has certain associations with parameters as variable content and therefore it cannot be acceptable in every case.
Factors that are to be taken into consideration:
While dealing with queuing theory, there are a number of factors that needs to be taken into consideration.
- Exponential distribution:
As one of the most commonly used format in distribution of queuing theory, this specifically deals with probability of completion of a service and ways to reduce costs associated with queuing theory. In case of a queue, there are costs associated with its such as waiting cost and service providing cost, and it is the aim of this theory to reduce these costs to garner maximum profit.
It is here a queue model is required which is prepared by taking into consideration different variables. With no maximisation or minimisation attempted, this model can be worked upon with help of different alternatives.
- Service pattern:
This is known as the rate of arrival of customers and can be differentiatedon the basis of haphazard or patterned arrival. It is assumed that service patterns are exponential to ensure that complex mathematical values can be avoided at all costs.
- Various channels of service:
In this case two types of channels can be taken into consideration, the single channel format and the service in series format. For different customers and service centres the options are different and as and when required.
- Series related service: In this case, input that has been given to a particular domain moves from one service centre to another in a parallel manner. This is how a raw material is processed in the industry, from one step to another.
- Single channel system: In this case, there is a single service window where customers arrive and wait for their turn to get served. Once they are rendered the required service, they leave the queue and make way for the next person in line.
- Multi-parallel service with a single queue system: In this case, there are multiple options provided against a single queue. Standing at a single queue people can get multiple services rendered to them as soon as the reach the peak point.
- Multi-parallel services with multiple queues system:In case of this system, there is more than a single queue and more than one service station to serve the people concerned. Each queue has separate facilities to attend to specific customers.
- Erlang Distribution:
To make the concept of exponential distribution simple, it is assumed that in the queuing process, service is either consistent or follows a negative exponential making the values of sigma equal to mean value.
However, in certain cases, these values are not the same and in those areas, it is imperative that one makes use of Erlang Distribution. In comparison to other formats, here it is assumed that service time is divided into a number of phases where totality of phases is taken into consideration. The exponential distribution is followed by the service time that is taken into consideration.
- Service time:
This is specifically dependent on demands of the consumer and the manner in which he or she is served. Since arrival of customers is in a haphazard mode hence, it is evident that demands of every customer is different and immediate changes need to be made by the service centre to adhere to demands of that particular client.
In such a scenario, to make this process easier, it is assumed that time required by each of the customers is constant.
- Rate of traffic and it’s utilisation:
This is the rate at which service facility is utilised by components ton whom it is dedicated.
Here, λ = mean arrival rate, (Mue) μ = Mean service rate, utilisation rate (P) = λ/ μ
- Idle rate:
This is the rate at which the service facility is not being used for any purpose.
Certain denominations are:
Expected number of customers: Customers standing in queue + Customers being served.
Average length of queue: Number of expected customers – Number of customers being served.
Expected time spent: Time spent in waiting + Time spent in servicing.
When probability of a customer wait is zero, it is taken that no customer would be waiting in the line to be served.
Queuing discipline:
When all the customers get into the queue and then are served, there are certain standards followed for serving them. It is these techniques that are known as queuing discipline.
- Service in Priority: This is a system where people having the highest priority and especially those in sensitive areas get the required service.
- Last come first serve: This is a system where people who come last are served at the first instance depending on their requirement.
- First come first serve: This is the usual technique that is followed, wherein people who have come first are served immediately.
Queuing cost behaviour:
This is known as the total cost that a service provider incurs in regards to serving people who are standing in the queue and those who are actuallybeing served.
Customer behaviour:
- The Collusion format shows those customers who do not come themselves but rather send representatives to attend the queue on their behalf.
- The Jockeying process is one when customer leaves one queue that is snot serving fast and joins another queue to get immediate service.
- Reneging is a system where customers who do not have patience leave the queue without being served.
- Balking is a system where people waiting in the queue jump the queue to get served earlier.
Thus, for a complete knowledge of the queuing theory, detailed understanding is of prime importance.
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- Duality in linear programming
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- Limitations of queuing theory
- Important terms used in queuing theory
- Types of queuing models
- Single channel queuing model
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