Queuing Theory Notes

KRUPANIDHI SCHOOL OF MANAGEMENT

PROF. RAMALAKSHMI.V Page 1

Queuing Theory Notes

1. Define Queuing theory.

The Queuing theory is the probabilistic study of waiting lines. Although it does not

solve all types of waiting line problems, yet it provides useful and vital information by

forecasting or predicting the various characteristics and parameters of the particular

waiting line under study. Since the prediction about the waiting times, the number of

waitings at any time, the time for which the server/servers remain busy etc. rely heavily

on the basic concept of stochastic processes, it can very well be taken as an application of

stochastic processes. Also, queuing theory is generally considered a branch of operational

research because the results are often used when making decisions about the resources

needed to provide service. Thus, queuing theory is an application of stochastic

processes in O.R.

2. Define queue

Gathering of some people or things at some place for some purpose is called queue.

3. Define queuing system.

A queuing system can be described as customers arriving for service, waiting for service

if it is not immediate, and if having waited for service, leaving the system after being

served.

Examples: Telephone conversation, Landing of Aircraft, Ticket windows, Taxi Stands,

Loading and Unloading of Ships, Scheduling patients in hospital clinics

Applications in the computer field with respect to programscheduling, time-sharing, and

system design.

4. Explain the terminologies of queuing theory.

Customers independent entities that arrive at random times to a Server and wait for some

kind of service, then leave.

Server can only service one customer or a batch at a time; length of time to provide

service depends on type of service.

Queue Length at time t number of customers in the queue at time t.

Waiting Time for a given customer, how long that customer has to wait between arriving

at the server and when the server actually starts the service (total time is waiting time plus

service time).



5. What are the characteristics of queuing theory?

Arrival pattern of customers.

Service pattern of customers.

Queue discipline.

System capacity.

Number of service channels.

Number of Service stages.

6. Explain the Arrival Pattern of Customers.

The arrival pattern or input to a queuing system is often measured in terms of average

number of arrivals per some unit of time or by the average time between successive

arrivals.

Arrival can occur either one by one or in batches.

If stream of input is deterministic, then the arrival pattern is fully determined by

either the mean arrival order or the mean inter-arrival time.

If arrival pattern is uncertain or random, then these mean values provide only

measures of central-tendency for the input process and further characterization is

required in the form of the probability distribution associated with this random

process. (In business world customers arrive in no logical-pattern or order over

time.)

Customers Behavior:- It also depends on customers behavior.

Balked:- After seeing the queue customers decides not to enter the queue.

Reneged:- Customers may enter the queue, but after time lose patience they decide to

leave.

Jockey:- In Case when two or more parallel queues, customers may switch from one

to another.

7. Explain the Service Patterns of Servers.

Service patterns can also be described by a rate (no. of customers served per unit of time)

or as a time (time required to service a customer).

If the system is empty, the service facility is idle.

Service may also be deterministic or probabilistic; hence in latter case the probability

distributions associated with service are conditional, based on the nonempty system.

Service rate may depend on the no. of customers waiting for service. A server may work

faster if he sees that the queue is building up or he may get frustrated and become less

efficient. This type of situation is referred to as state-dependent service.



8. What is Queue Discipline?

FIFO- first in first out

LIFO- last in first out

SIRO-service in random order

Service in priority

preemptive ( very high priority)

no preemptive

9. What is System Capacity?

In some queuing processes there is a physical limitation to the amount of waiting room,

so that when the line reaches a certain length, no further customers are allowed to enter

until space becomes available by a service completion. These are referred to as finite

queuing situations.

10. Explain Stages of Service.

A queuing system may have only a single stage of service such as the barber shop and

supermarket examples, or it may have several stages. An example of multistage queuing

systems would be a physical examination procedure, where patient must proceed through

several stages, such as medical history; ear, nose, and throat examination; blood tests;

electro- cardiogram; eye examination; and so on.

11. What is M/M/1 Queues?

1st M (for Markovian) Arrival Distribution is Exponential

2nd

M Service Distribution is Exponential

1 Single Channel

12. What is Arrival Process?

The inter-arrival time is an exponentially-distributed random variable with average

arrival rate = .

If the inter-arrival time is an exponentially-distributed random variable, then the number

of arrivals during the fixed period of time is a Poisson distribution.

No balking or reneging

13. What is Service Process?

The service time is also assumed to be exponentially distributed with mean service rate .

Only 1 server



First-come-first-served (FCFS) queue priority

Mean length of service = 1/

No limit on the queue size.

14. Explain the Operating Characteristics of Queuing Systems

Operational characteristics of queuing system, that are of a general interest for the

evaluation of the performance of an existing queuing system and to design a new system

are as follows:

Expected No. of Customers in the system denoted by L is the average no. of customers

in the system, both waiting and in service. Here, n stands for the no. of customers in the

queuing system.

Expected waiting time in the system is denoted by W is the average total time spent by a

customer in the system. It is generally taken to be the waiting time plus servicing time.

Expected no. of customers in the queue denoted by Lq is the average no. of customers in

the queue. Here m=n-1, i.e., excluding the customer being served.

Expected waiting time in queue denoted by Wq is the average time spent by a customer

in the queue before the commencement of his service.

Expected waiting time in queue denoted by Wq is the average time spent by a customer

in the queue before the commencement of his service.

The service utilization factor (Traffic Intensity) denoted by (lambda/mu) is the

proportion of time that a server actually spends with the customers( or it determines

the degree to which the capacity of the service station or server is utilized) .

Here, lambda stands for the average no. of customers arriving per unit of time and mu

stands for the average no. of customers completing service per unit of time.

Since most queuing systems have stochastic elements, these measures are often random

variables and there probability distributions are desired to be found.

The task of the queuing analyst is one of two things. He must either determine the

values of appropriate measures of effectiveness for a given process or he must design

an optimal system (according to some criterion).

TO do the former he must relate waiting delays, queue lengths, to the given properties of

the input stream and the service procedures.



On the other hand, for the design of a system the analyst would probably want to balance

customer waiting time against the idle time of the serves according to some inherent

cost structure. If the cost of waiting and idle service can be obtained directly, they could

be used to determine the optimum number of channels to maintain and the service rates at

which to operate these channels.

Also, to design the waiting facility it is necessary to have information regarding the

possible size of the queue to plan for waiting room. There may also be space cost which

should be considered along with customer-waiting and idle-server cost to obtain the

optimal system design.

A stable system: The queue will never increase to infinity. An empty state is reached for

sure after some time period.

Condition for Stability: >. This condition MUST be met to make all formulas valid.

The steady state: Probability {n customers in the system} does not depend on the time.

15. Explain the classification of queuing models

Kendall notation Parameter Description

(M/M/1):(GD//) P Poisson arrival rate

Q Poisson service rate

R Single server

X General discipline

Y Infinite number of customers is permitted in the system

Z Size of the calling source is infinite

(M/M/C):(GD//) P Poisson arrival rate


R Multiservers


Y infinite number of customers is permitted in the system


(M/M/1):(GD/N/) P Poisson arrival rate


R Single server


Y Finite number of customers is permitted in the system


(M/M/C):(GD/N/) P Poisson arrival rate




R Multi servers




(M/M/1):(GD/N/N) P Poisson arrival rate


R Single server



Z Size of the calling source is finite

(M/M/C):(GD/N/N) P Poisson arrival rate


R Multiservers




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Queuing Theory Notes