4Queuing_Theory.ppt
- 2. What is queuing theory? • Queuing Theory is a collection of mathematical models of various queuing systems. It is used extensively to analyze production and service processes exhibiting random variability in market demand (arrival times) and service times.
- 3. Why queues form? • Queues or waiting lines arise when the demand for a service facility exceeds the capacity of that facility, that is, the customers do not get service immediately upon request but must wait, or the service facilities stand idle and wait for customers. • Most waiting line problems are centered about the question of finding the ideal level of services that a firm should provide.
- 4. For example • Supermarkets must decide how many cash register check out positions should be opened. • Manufacturing plants must determine the optimal number of mechanics to have on duty in each shift to repair machines that break down. • Banks must decide how many teller windows to keep open to serve customers during various hours of the day.
- 6. BASIC COMPONENTS OF A QUEUING SYSTEM • INPUT SOURCE OF QUEUE An input source is characterized by • Size of the calling population • Pattern of arrivals at the system • Behaviour of the arrivals
- 7. Size of the calling population The size represents the total number of potential customers who will require service. According to source • The source of customers can be finite or infinite. For example, all people of a city or state (and others) could be the potential customers at a supermarket. The number of people being very large, it can be taken to be infinite. Whereas there are many situations in business and industrial conditions where we cannot consider the population to be infinite—it is finite. According to numbers • The customers may arrive for service individually or in groups. Single arrivals are illustrated by patients visiting a doctor, students reaching at a library counter etc. On the other hand, families visiting restaurants, ships discharging cargo at a dock are examples of bulk, or batch arrivals.
- 8. According to time • Customers arrive in the system at a service facility according to some known schedule (for example one patient every 15 minutes or a candidate for interview every half hour) or else they arrive randomly. Arrivals are considered random when they are independent of one another and their occurrence cannot be predicted exactly. The queuing models wherein customers’ arrival times are known with certainity are categorized as deterministic models. (insofar as this characteristic is concerned) and are easier to handle. On the other hand, a substantial majority of the queuing models are based on the premise that the customers enter the system stochastically, at random points in time.
- 9. Pattern of arrivals at the system • The arrival process (or pattern) of customers to the service system is classified into two categories: • Static • Dynamic. • These two are further classified based on the nature of arrival rate and the control that can be exercised on the arrival process. – In static arrival process, the control depends on the nature of arrival rate (random or constant). Random arrivals are either at a constant rate or varying with time. Thus to analyze the queuing system, it is necessary to attempt to describe the probability distribution of arrivals. From such distributions we obtain average time between successive arrivals, also called inter-arrival time (time between two consecutive arrivals), and the average arrival rate (i.e. number of customers arriving per unit of time at the service system). – The dynamic arrival process is controlled by both service facility and customers. The service facility adjusts its capacity to match changes in the demand intensity, by either varying the staffing levels at different timings of service, varying service charges (such as telephone call charges at different hours of the day or week) at different timings, or allowing entry with appointments. • Frequently in queuing problems, the number of arrivals per unit of time can be estimated by a probability distribution known as the Poisson distribution, as it adequately supports many real world situations.
- 10. Behavior of arrivals • On the basis of the behavior or attitude of the customers entering the queuing system, the customers may be classified as being (a) patient, or (b) impatient.
- 11. Some interesting observations of human behavior in queues : • Balking – Some customers even before joining the queue get discouraged by seeing the number of customers already in service system or estimating the excessive waiting time for desired service, decide to return for service at a later time. In queuing theory this is known as balking. • Reneging - customers after joining the queue, wait for sometime and leave the service system due to intolerable delay, so they renege. – For example, a customer who has just arrived at a grocery store and finds that the salesmen are busy in serving the customers already in the system, will either wait for service till his patience is exhausted or estimates that his waiting time may be excessive and so leaves immediately to seek service elsewhere. • Jockeying - Customers who switch from one queue to another hoping to receive service more quickly are said to be jockeying.
- 12. SERVICE SYSTEM • The service is provided by a service facility (or facilities). This may be a person (a bank teller, a barber, a machine (elevator, gasoline pump), or a space (airport runway, parking lot, hospital bed), to mention just a few. A service facility may include one person or several people operating as a team. • There are two aspects of a service system— (a) the configuration of the service system and (b) the speed of the service.
- 13. (a) The configuration of the service system • The customers’ entry into the service system depends upon the queue conditions. If at the time of customers’ arrival, the server is idle, then the customer is served immediately. Otherwise the customer is asked to join the queue, which can have several configurations. • By configuration of the service system we mean how the service facilities exist. Service systems are usually classified in terms of their number of channels, or numbers of servers
- 14. Single Server – Single Queue The models that involve one queue – one service station facility are called single server models where customer waits till the service point is ready to take him for servicing. Students arriving at a library counter is an example of a single server facility. • Arrivals Queue Service Facility Customers Leave
- 15. Single Server – Several Queues • In this type of facility there are several queues and the customer may join any one of these but there is only one service channel.
- 17. Several (Parallel) Servers – Single Queue • In this type of model there is more than one server and each server provides the same type of facility. The customers wait in a single queue until one of the service channels is ready to take them in for servicing.
- 19. Several Servers – Several Queues • This type of model consists of several servers where each of the servers has a different queue. Different cash counters in an electricity office where the customers can make payment in respect of their electricity bills provide an example of this type of model.
- 20. Service facilities in a series • In this, a customer enters the first station and gets a portion of service and then moves on to the next station, gets some service and then again moves on to the next station. …. and so on, and finally leaves the system, having received the complete service. • For example, machining of a certain steel item may consist of cutting, turning, knurling, drilling, grinding, and packaging operations, each of which is performed by a single server in a series.
- 21. b) Speed of Service • In a queuing system, the speed with which service is provided can be expressed in either of two ways—as service rate and as service time. • The service rate describes the number of customers serviced during a particular time period. • The service time indicates the amount of time needed to service a customer. • Service rates and times are reciprocal of each other and either of them is sufficient to indicate the capacity of the facility. Generally, we consider the service time only.
- 22. QUEUE CONFIGURATION • The queuing process refers to the number of queues, and their respective lengths. The number of queues depend upon the layout of a service system. Thus there may be a single queue or multiple queues. – Length (or size) of the queue depends upon the operational situation such as • physical space, • legal restrictions, and • attitude of the customers.
- 23. QUEUE DISCIPLINE • The queue discipline is the order or manner in which customers from the queue are selected for service. (a) Static queue disciplines are based on the individual customer's status in the queue. Few of such disciplines are: – i If the customers are served in the order of their arrival, then this is known as the first-come, first-served (FCFS) service discipline. Prepaid taxi queue at airports where a taxi is engaged on a first-come, first-served basis is an example of this discipline. – ii Last-come-first-served (LCFS)-- Sometimes, the customers are serviced in the reverse order of their entry so that the ones who join the last are served first. For example, assume that letters to be typed, or order forms to be processed accumulate in a pile, each new addition being put on the top of them. The typist or the clerk might process these letters or orders by taking each new task from the top of the pile. Thus, a just arriving task would be the next to be serviced provided that no fresh task arrives before it is picked up. Similarly, the people who join an elevator last are the first ones to leave it. • .
- 24. • (b) Dynamic queue disciplines are based on the individual customer attributes in the queue. Few of such disciplines are: • i Service in Random Order (SIRO)-- Under this rule customers are selected for service at random, irrespective of their arrivals in the service system. In this every customer in the queue is equally likely to be selected. The time of arrival of the customers is, therefore, of no relevance in such a case. ii Priority Service-- Under this rule customers are grouped in priority classes on the basis of some attributes such as service time or urgency or according to some identifiable characteristic, and FCFS rule is used within each class to provide service. Treatment of VIPs in preference to other patients in a hospital is an example of priority service. • For the queuing models that we shall consider, the assumption would be that the customers are serviced on the first-come-first-served basis