Theory of queuing systems
- 1. 1 Theory of QueuingSystems Waiting for service is an integralpart of modern life. We wait for buying tickets at railway, bus or metro ticket counters, to board airplane or to enter into a movie theatre. Waiting is not only a phenomenon limited within the sphere of human experience. In factories, jobs wait to be attended by machines, materials wait to be inventoried and coded. Such experiences of waiting happen because in each of the cases, there is congestion of different items arriving for service. Such congestion occurs because an item cannot be served immediately on arrival and each new arrival has to wait for some time before it is attended. Absolute elimination of waiting altogether in a service facility or in any case is never a feasible option as it involves the attending all the items at the same time with identicalprocess time. Even if such a facility is a realistic option, the cost and other issues involved would make it impossible. Hence, waiting will be there and queues will be formed. The above discussion leads towards the formation of a definition of queue: A group of items waiting to receive service, including those receiving the service, is known as queue. Queues cannot be eliminated completely. However, it is possible to strike a balance between the cost of the service and the cost of the waiting. The study of queues deals with quantifying the phenomenon ofwaiting by applying various measures of performance, such as the average length of the queue, average waiting time in queue and average facility utilization. The study of the theory of queuing system helps to determine the balance between the i) cost of offering the service and ii) cost of waiting for the service. Cost-based Model: The analysis of the queuing system may be incorporated in a cost optimization mode that aims at the minimization of the sum of the cost of offering the service and the cost of waiting for the service. The figure below presents a typicalcost model. In the presented model,the cost of service increases with the increase in the level of service and the cost of waiting decreases with the increase in the level of service.
- 2. 2 Figure-1: Cost-based queuing decision model The results obtained from queuing analysis may be obtained in a cost optimization mode that tends to minimize the sum of the i) cost of operating the service facility and the ii) cost of waiting by customers. Figure-1 above represents the typical cost model. As observed in the figure, the cost of operating the service facility increases exponentially with the increase in the level of service. On the other hand, the cost of waiting decreases with the increase in the level of service. The curves intersect each other at a certain point, and the projection of this point on the X axis represents the optimum level of service. The corresponding total cost is the parabola, with its lowest point presenting the optimum cost and optimum level of service. The Features of the Structure of a Queuing System A queuing system is composed of a number of components as given below: 1. Input Source or calling population 2. Queuing process 3. Queue Discipline 4. Service Process or mechanism Cost of waiting customers per unit time Cost Total Cost Cos t of operating the s ervice facility per unit time Optimum Level of Service Level of Service
- 3. 3 Input Sourceor calling population: Customers in a queuing system are defined by the type of services they require. Customers, coming to join a queuing system expecting some service are called calling population or input source. An input source is characterized by a) its size, b) the arrival time distribution of the customers and c) the attitude of the customers. a) Size: A calling population may be infinite or finite in numbers. If the rate of generation ofnumberof customers by the source is not affected by the number of customers in the service system the calling population is infinite. Example: On the otherhand,if the rate of generation ofnumber of customers by the source is affected by the numberof customers in the service system the calling population is considered finite. Input Source or calling population Queue or waiting line Service process or mechanism Balk Arrival Process Waiting Area Queue Discipline Renege Jockey Service Facility Departure