Queuing Systems
Operations Management
Queuing Systems
Waiting for service is part of daily life. We wait for service in restaurants, we queue up to board a plane, and we line up for service in post offices.
And the waiting phenomenon is not an experience limited to human beings: Jobs wait to be processed on a machine, planes circle in stack before given permission to land, and cars stop at traffic lights.
Queuing Systems
A fast-food restaurant with three service counters. The manager wants to expedite service. A study reveals the following relationship between the number of service counters and the waiting time for service:
An examination of these data shows a 7-min average waiting time for the present 3-counter situation. Five counters would reduce waiting to about 3 minutes.
Elements of a Queuing Model
The principal players in a queuing situation are the customer and the server. Customer arrive at a (service) facility from a source. On arrival, a customer can start service immediately or wait in a queue if the facility is busy.
When a facility completes a service, it automatically “pulls” a waiting customer, if any, from the queue. If the queue is empty, the facility becomes idle until a new customer arrives.
Elements of a Queuing Model
From the standpoint of analyzing queues, the arrival of customers is represented by the interarrival time (time between successive arrivals), and the service is measured by the service time per customer.
The interarrival and service time are:
Elements of a Queuing Model
Queue size plays a role in the analysis of queues, it may be:
Queue discipline represents the order in which customer are selected from a queue, is an important factor in the analysis of queuing models.
Customers may also be selected from the queue based on some order of priority.
Elements of a Queuing Model
Queuing behavior plays a role in waiting-line analysis. Customers may jockey from a longer queue to a shorter one to reduce waiting time, they may balk from joining a queue altogether because of anticipated long delay, or they may renege from a queue because they have been waiting too long.
The design of the service facility may include parallel servers (e.g., post office or bank operation). The servers may also be arranged in series (e.g., jobs processed on successive machines), or they may be networked (e.g., routers in a computer network).
Elements of a Queuing Model
The source from which customers are generated may be finite or infinite. A finite source limits the number of arriving customers (e.g., machines requesting the service of a repairperson). An infinite source is, for all practical purposes, forever abundant (e.g., calls arriving at a telephone exchange).
Role of Exponential Distribution
In most queuing situations, arrivals occur randomly. Randomness means that the occurrence of an event (e.g., arrival of a customer or completion of a service) is not influenced by the length of time that has elapsed since the occurrence of the last event.
Random interarrival and service times are described quantitatively in queuing models by the exponential distribution, which is defined as
The exponential distribution describes a totally random phenomenon.
Poisson Distribution
When we are interested in the number of arrivals or events in a period T, the distribution is obtained by finding the exact probability of n arrivals during T.
If the arrival process is random, then the distribution is Poisson, and the formula is: