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UNIT – III Definition of queuing theory: Mathematical modeling of waiting lines, whether of people, signals, or things. It aims to estimate if the available resources will suffice in meeting the anticipated demand over a given period. Queuing theory Queuing theory deals with problems which involve queuing (or waiting). Typical examples might be: banks/supermarkets - waiting for service computers - waiting for a response failure situations - waiting for a failure to occur e.g. in a piece of machinery public transport - waiting for a train or a bus As we know queues are a common every-day experience. Queues form because resources are limited. In fact it makes economic sense to have queues. For example how many supermarket tills you would need to avoid queuing? How many buses or trains would be needed if queues were to be avoided/eliminated? In designing queueing systems we need to aim for a balance between service to customers (short queues implying many servers) and economic considerations (not too many servers). In essence all queuing systems can be broken down into individual sub-systems consisting of entities queuing for some activity (as shown below). Typically we can talk of this individual sub-system as dealing with customers queuing for service. To analyse this sub-system we need information relating to:
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Page 1: Queuing theory - snscourseware.org€¦  · Web viewQueuing theory. Queuing theory deals with problems which involve queuing (or waiting). Typical examples might be: banks/supermarkets

UNIT – IIIDefinition of queuing theory:

Mathematical modeling of waiting lines, whether of people, signals, or things. It aims to estimate if the available resources will suffice in meeting the anticipated demand over a given period.

Queuing theory

Queuing theory deals with problems which involve queuing (or waiting). Typical examples might be:

banks/supermarkets - waiting for service computers - waiting for a response failure situations - waiting for a failure to occur e.g. in a piece of machinery public transport - waiting for a train or a bus

As we know queues are a common every-day experience. Queues form because resources are limited. In fact it makes economic sense to have queues. For example how many supermarket tills you would need to avoid queuing? How many buses or trains would be needed if queues were to be avoided/eliminated?

In designing queueing systems we need to aim for a balance between service to customers (short queues implying many servers) and economic considerations (not too many servers).

In essence all queuing systems can be broken down into individual sub-systems consisting of entities queuing for some activity (as shown below).

Typically we can talk of this individual sub-system as dealing with customers queuing for service. To analyse this sub-system we need information relating to:

arrival process:o how customers arrive e.g. singly or in groups (batch or bulk arrivals)o how the arrivals are distributed in time (e.g. what is the probability distribution

of time between successive arrivals (the interarrival time distribution))o whether there is a finite population of customers or (effectively) an infinite

number

The simplest arrival process is one where we have completely regular arrivals (i.e. the same constant time interval between successive arrivals). A Poisson stream of arrivals corresponds to arrivals at random. In a Poisson stream

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successive customers arrive after intervals which independently are exponentially distributed. The Poisson stream is important as it is a convenient mathematical model of many real life queuing systems and is described by a single parameter - the average arrival rate. Other important arrival processes are scheduled arrivals; batch arrivals; and time dependent arrival rates (i.e. the arrival rate varies according to the time of day).

service mechanism:o a description of the resources needed for service to begino how long the service will take (the service time distribution)o the number of servers availableo whether the servers are in series (each server has a separate queue) or in

parallel (one queue for all servers)o whether preemption is allowed (a server can stop processing a customer to

deal with another "emergency" customer)

Assuming that the service times for customers are independent and do not depend upon the arrival process is common. Another common assumption about service times is that they are exponentially distributed.

queue characteristics:o how, from the set of customers waiting for service, do we choose the one to be

served next (e.g. FIFO (first-in first-out) - also known as FCFS (first-come first served); LIFO (last-in first-out); randomly) (this is often called the queue discipline)

o do we have: balking (customers deciding not to join the queue if it is too long) reneging (customers leave the queue if they have waited too long for

service) jockeying (customers switch between queues if they think they will get

served faster by so doing) a queue of finite capacity or (effectively) of infinite capacity

QUEING SYSTEM:

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Definition: Structure of Waiting Line ModelWaiting line model consists of various parameters like cost of waiting (Cw), cost of service (Ck), time taken for an object in the waiting line (Wq), time taken for an object in the entire system (W), average number of units in the system (L), average number of unit in waiting line (Lw). The rate of service is directly proportional to the cost of service while the cost of waiting is inversely proportional to the rate of service.

It means if the cost of service is high, the cost of waiting will be low while the rate of service will be high. The Waiting line model stipulates that the rate of service should be at an optimal level in order for doctors, lawyers and others who have a high price for time.The rate of service is directly proportional to the cost of service while the cost of waiting is inversely proportional to the rate of service.

It means if the cost of service is high, the cost of waiting will be low while the rate of service will be high. The Waiting line model stipulates that the rate of service should be at an optimal level in order for doctors, lawyers and others who have a high price for time.

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Arrival ProcessArrivals to the waiting line system from the population source may be on an individual or batch basis. We will assume arrivals are on an individual basis. The difference is best illustrated by the arrival of a car to a parking lot at a restaurant. One driver leaving the car to enter the restaurant would represent the arrival of one unit or customer to the waiting line system. If a bus pulls in, there could be a batch arrival of 30 customers. Did you ever notice that the bus stalls are behind the Cracker Barrel Restaurants on the interstate highways - just so you can't see all those batch arrivals before you pull off!

It is also assumed that the arrivals are nonscheduled, and the arrival of one unit is independent of, or does not impact, the arrival of other units. Whenever these assumptions are made, arrivals are assumed to follow the Poisson Probability Distribution, a member of the family of discrete probability distributions. The Poisson Probability Distribution is completely described by its mean, which is given the Greek symbol lambda. In a waiting line system, the mean we are referring to is the mean arrival rate. For example, we may say that the mean arrival rate is 4 calls per hour to a catalog company's telephone bank.

Another way of representing the mean arrival rate is to take its inverse, which gives us the mean time between arrivals. So, if I invert the mean rate of 4 calls per hour, I get ¼ hours. The mean time between arrivals is 1/4th of an hour, or one arrival every 15 minutes.

Mean Time Between Arrivals = 1 / Mean Arrival Rate

The probability distribution that is used to describe this time between arrivals in a waiting line system is the Exponential Distribution. The Exponential Distribution is used to model the probabilities of continuous variables such as time, in the case of waiting line systems. The Greek Symbol Mu, is used to describe the mean of the Exponential Distribution.

If you have the mean time between arrivals, you can find the mean arrival rate by the similar procedure - taking the inverse. For example, what if we knew that the average time between arrivals to a bank teller was 5 minutes. The mean arrival rate would be computed as follows:

Mean Arrival Time = 5 minutes = 5/60th hours

Mean Arrival Rate = 60/5 = 12 customers per hour

Customers arriving from a population next join the waiting line in the waiting line system.

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Total Time in System

• Service time: the amount of time you would be delayed if no other customers required service

• Waiting time: the amount of time you have to wait because others also want service– The price you pay for others

• Total Time in System = Service time + Waiting time

Queue Discipline

• FIFO– Traffic– intersection

• LIFO– Elevator– Airplane

• Random– Fluids

• Priority

TRAFFIC INTENCY:

In telecommunication networks, traffic intensity is a measure of the average occupancy of a server or resource during a specified period of time, normally a busy hour. It is measured in traffic units (erlangs) and defined as the ratio of the time during which a facility is cumulatively occupied to the time this facility is available for occupancy.

In a digital network, the traffic intensity is:

where

a is the average arrival rate of packets (e.g. in packets per second)L is the average packet length (e.g. in bits), andR is the transmission rate (e.g. bits per second)

A traffic intensity greater than one erlang means that the rate at which bits arrive exceeds the rate bits can be transmitted and queuing delay will grow without bound (if the traffic intensity stays the same). If the traffic intensity is less than one erlang, then the router can handle more average traffic.

Telecommunication operators are vitally interested in traffic intensity, as it dictates the amount of equipment they must supply.

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Population:

The word population or statistical population is used for all the individuals or objects on which we have to make some study. We may be interested to know the quality of bulbs produced in a factory. The entire product of the factory in a certain period is called a population. We may be interested in the level of education in primary schools. All the children in the primary schools will make a population. The population may contain living or non-living things.

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The entire lot of anything under study is called population. All the fruit trees in a garden, all the patients in a hospital and all the cattle in a cattle form are examples of population in different studies.

Finite Population:A population is called finite if it is possible to count its individuals. It may also be called a countable population. The number of vehicles crossing a bridge every day, the number of births per years and the number of words in a book are finite populations. The number of units in a finite population is denoted by NN. Thus NN is the size of the population

Infinite Population:Sometimes it is not possible to count the units contained in the population. Such a population is called infinite or uncountable. Let us suppose that we want to examine whether a coin is true or not. We shall toss it a very large number of times to observe the number of heads. All the tosses will make an infinite or countable infinite population. The number of germs in the body of a patient of malaria is perhaps something which is uncountable.


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