System Engineering 3(3+0)

Lesson 2 CHARACTERISTICS OF QUEUEING SYSTEMS

The Input Process. It expresses the mode of arrival of customers at the service facility governed by some probability law. The number of customers emanate from finite or infinite sources. Also, the customers may arrive at the service facility in batches of fixed size or of variable size or one by one. In the case when more than one arrival is allowed to enter the system simultaneously, (entering the system does not necessarily mean entering into service), the input is said to occur in bulk or batches.

 

It is also necessary to know the reaction of a customer upon entering the system. A customer may decide to wait no matter how long the queue becomes, or if the queue is too long to suit him, may decide not to enter it. If a customer decides not to enter the queue because of its huge length, he is said to have balked. On the other hand, a customer may enter the queue, but after some time loses patience and decides to leave. In this case he is said to have reneged. In the case when there are two or more parallel queues, the customer may move from one queue to another for his personal economic gains, that is jockey for position.

 

The final factor to be considered regarding the input process is the manner in which the arrival pattern changes with time. The input process which does not change with time is called a stationary input process. If it is time dependent then the process is termed as transient.

 

The Queue Disline. cipIt is a rule according to which customers are selected for service when a queue has been formed. The most common discipline is the "first come, first served" (FCFS), or "first in, first out" (FIFO) rule under which the customers are serviced in the strict order of their arrivals. Other queue disciplines include: "last in, first out" (LIFO) rule according to which the last arrival in the system is serviced first, "selection for service in random order" (SIRO) rule according to which the arrivals as serviced randomly irrespective of their arrivals in the system; and a variety of priority schemes-according to which a customer's service is done in preference over some other customer's service.

 

Under priority discipline, the service is of two types. In the first, which is called pre-emptive, the customers of high priority are given service over the low priority customer. In the second type, called the non-pre-emptive, a customer of low priority is serviced before a customer of high priority is entertained for service.

 

In the case of parallel channels "fastest server rule" (FSR) is adopted. For its discussion we suppose that the customers arrive before parallel service channels. If only one service channel is free, then incoming customer is assigned to free service channel. But it will be more efficient to assume that an incoming customer is to be assigned a server of largest service rate among the free ones.

The Service Mechanism. This means the arrangement of server-s facility to serve the customers. If there are infinite numbers of servers then all the customers are served instantaneously on arrival and there will be no queue.

 

 If the number of servers is finite, then the customers are served according to a specific order. Further, the customers may be served in batches of fixed size or of variable size rather than individually by the same server, such as a computer with parallel processing or people boarding a bus. The service system in this case is termed as bulk service system.

 

Sometimes, the service rate may also depend on the number of customers, waiting for service. For example, when the queue becomes longer, a server may work faster or, conversely, may become less efficient. The situation in which service depends upon the number of waiting customers is referred to as state dependent-system.

 

The Capacity of the System.  Some of the queueing processes admit the physical limitation to the amount of waiting room, so that when the waiting line reaches a certain length, no further customers are allowed to enter until space becomes available by a service completion. Such types of situation are referred to as finite source queues, that is, there is a finite limit to the maximum queue size. The queue can also be viewed as one with forced balking

Where a customer is forced to balk if he arrives at a time when queue size is at its limit.       .

 

Service Channels: When there are several service channels available to provide service, much depends upon their arrangements. They may be arranged in parallel or in series or a more complex combination of both, depending on the design of the system's service mechanism.

 

By parallel channels we mean a number of channels providing identical service facilities so that several customers may be serviced simultaneously. Further, customers may wait in a single queue until one of the service channels is ready to serve, as in a barber shop where many chairs are considered as different service channels; or customers may form separate queues in front of each service channel as in the case of super markets.

 

For series channels, a customer must pass successively through all the ordered channels before service is completed. The situations may be seen in public offices where parts of the service are done at different service counters.

A queueing system is called a one-server model when the system has one server only, and a multiple-server model when the system has a number of parallel channels each with one server.

SYMBOLS AND NOTATIONS

 

The following symbols and notations will be used in connection with the queuing systems:

 

n = number of customers in the system, both waiting and in service.

λ = average number of customers arriving per unit of time.

 µ = average number of customer being served per unit of time.

λ / µ = Ρ = traffic intensity.

C = number of parallel service channels (servers).

E(n) = average number of customers in the system. Both waiting and in service.

 E(m) = average number of customers waiting in the queue.

E(v) = average waiting time of a customer in the system, both waiting and in service.

E(w) = average waiting time of a customer in the queue.

P_n(t) = Probability that there are n customers in the system at any time t, both waiting and in service.

P_n = time independent Probability that there are n customers in the system at any time, both waiting and in service.