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Important terms used in queuing theory


  1. 1.   Arrival Pattern. It is the pattern of the arrival of a customer to be serviced. The pattern may be regular or at random. Regular interval arrival patterns .are rare, in most of the cases of the customers cannot be predicted. Remainder pattern of arrival of customers follows Poisson’s distribution.
  2. 2.   Poisson’s DistributionIt is- discrete probability distribution which is used to determine the number of customers in a particular time: It involves allotting probability of occurrence of the arrival of a customer. Greek letter λ (lamda) is used to denote mean arrival rate. A special feature of the Poisson’s distribution is that its mean is equal to the variance. It can be represented with the notation as explained below.P(n) = Probability of n arrivals (customers)λ = Mean arrival ratee = Constant = 2.71828Notation ∟ or ! is called the factorial and it mean that∟n or n! = n(n-1) (n-2) (n-3) …. 2, 1

    Poisson’s distribution tables for different values of n and λ are available and can be used for solving problems where Poisson’s distribution is used. However, It has certain limitations because of which its used is restricted. It assumes that arrivals are random and independent of all other variables or other variables or parameters. Such can never be the case.

    1. 1.   Exponential Distribution. This is based on the probability of completion of a service and is the most commonly used distribution in queuing theory. In queuing theory, our effort is to minimise the total cost of queue and it includes cost of waiting and cost of providing service. A queue model
      is prepared by taking different variables into consideration. In this distribution system, no maximisation or minimization is attempted. Queue models with different alternatives are considered and the most suitable for a particular is attempted. Queue models with different alternatives are considered and the most suitable for a particular situation is selected.
    2. 2.   Service Pattern. We have seen that arrival pattern is random and poissons distribution can be used for use in queue model. Service pattern are assumed to be exponential for purpose of avoiding complex mathematical problem.
    3. 3.   Channels. A service system has a number of facilities positioned in a suitable manner. These could be

    (a)             Single Channel- Single Phase System. This is very simple system where all the customers wait in a single line in front of a single service facility and depart after service is provided, In a shop if there is only one person to attend to a customer is an example of the system.

    (a)             Service in series. Here the input gets serviced at one service station and then moves to second and or third and so on before going out. This is the case when a raw material input has to undergo a number of operations like cutting, turning drilling etc.



  3. Here the service can be provided at a number of points to one queue. This happens when in a grocery store, there are 3 persons servicing the same queue or a service station having more than one facility of washing cars.(a)             Multiple parallel facilities with multiple queue. Here there are a number of queues and separate facility to service each queue. Booking of tickets at railway stations, bus stands etc is a good example of this, This is shown in Fig 9.4.
    1. 6.   Service Time. Service time i.e., the time taken by the customer when the facility is dedicated to it for serving depends upon the requirement of the customer and what needs to be done as assessed by the facility provider. The arrival pattern is random so also is the service time required by different customers. For the sake of simplicity the time required by all the customers is considered constant under the distribution. If the assumption of exponential distribution is not valid. Erlang Distribution is applied to the queuing model.
    2. 7.   Erlang Distribution. It has been assumed in the queuing process we have seen that service is either constant or it follows negative exponential distribution in which case the standard deviation a (sigma) is equal to its mean. This assumption makes the use of the exponential distribution simple. However, in cases where and mean are not equal, Erlang distribution developed by AK Erlang is used. In this method, the service time is divided into number of phases assuming that total service can be provided by different phases of service. It is assumed that service time of each phase follows the exponential distribution i.e., σ= mean.
    3. 8.   Traffic Intensity or Utilisation Rate. This is the rate of at which the service facility is utilised by the components.

    If λ = mean arrival rate and

    (Mue) μ = Mean service rate, then utilisation rate (P) = λ/ μ it can be easily seen from the equation that p > 1 when arrival rate is more than the service rate and new arrivals will keep increasing the queue p < 1 means that service rate is more than the arrival rate and the waiting time will keep reducing as μ keeps increasing. This is true from the commonsense.

    1. 6.   Queuing Discipline. All the customers get into a queue on arrival and are then serviced. The order in which the customer is selected for servicing is known as queuing discipline. A number of systems are used to select the customer to be served. Some of these are:
      1. a.    First in First Served (FIFS). This is the most commonly used method and the customers are served in the order of their arrival.
      2. b.   Last in First Served (LIFS). This is rarely used as it will create controversies and ego problems amongst the customers. Anyone who comes first expects to be served first. It is used in store management, where it is convenient to issue the store last-received and is called (LIFO) i.e., Last In First Out.
      3. c.    Service In Priority (SIP). The priority in servicing is allotted based on the special requirement of a customer like a doctor may attend to a serious patient out of turn, so may be the case with a vital machine which has broken down. In such cases the customer being serviced may be put on hold and the priority customer attended to or the priority may be put on hold and the priority customer attended to or the priority may be on hold and the priority customer waits till the servicing of the customer already being serviced is over.
    2. 7.   Customer Behaviour. Different types of customers behave in different manner while they are waiting in queue, some of the patterns of behaviour are:
      1. a.    Collusion. Some customers who do not want to wait they make one customer as their representative and he represents a group of customers. Now only the representative waits in queue and not all members of the group.
      2. b.   Balking. When a customer does not wait to join the queue at the correct place which he warrants because of his arrival. They want to jump the queue and move ahead of others to reduce their waiting time in the queue. This behaviour is called balking.
      3. c.    Jockeying. This is the process of a customer leaving the queue which he had joined and goes and joins another queue to get advantage of being served earlier because the new queue has lesser customers ahead of him.
      4. d.   Reneging Some customers either do not have time to wait in queue for a long time or they do not have the patience to wait, they leave the queue without being served.
    3. 8.   Queuing Cost Behaviour. The total cost a service provider system incurs is the sum of cost of providing the services and the cost of waiting of the customers. Suppose the garage owner wants to install another car washing facility so that the waiting time of the customer is reduced. He has to manage a suitable compromise in his best interest. If the cost of adding another facility is more than offset by reducing the customer waiting time and hence getting more customers, it is definitely worth it. The relationship between these two costs is shown below.