A group of items waiting to receive service, including those receiving a service is known as a waiting line or queue. Queuing theory involves the mathematical study of queues or waiting linesThe formation of waiting lines is a common phenomenon which occurs whenever the current demand for a service exceeds the current capacity to provide that service.

The queuing theory, also called the waiting line theory, owes its development to A K Erlangs effort to analyze telephone traffic congestion with a view to satisfy the randomly arising demand for the services of the Copenhagen automatic system, in the year 1909.

Banking transactionsPassage of customers through a supermarket checkoutCinema ticket windowReservation officeDoctors ClinicRegistration of unemployed at employment exchange

Balking. A customer may not like to wait in a queue due to lack of time or space or otherwise.Reneging. A customer may leave the queue due to impatienceCollusion. Some customers may collaborate and only one of them may join the queue and purchase tickets for his friendsJockeying. If there are more than one queues then one customer may leave one queue and join the other. This occurs generally in supermarket

The inputQueueThe service discipline (Service system )The service Mechanism (Service Pattern)

SERVICE FACILITY(How customers progress through the service facility)

Balking Customers

Reneging Customers

Waiting Line Models

The input describes the pattern in which customers arrive for service. Since the servicing units (customers) arrive in a random fashion, therefore their arrival pattern can be described in terms of probabilities. Here , we assume that they arrive according to a Poisson process i.e., the number of units arriving until any specific time has a Poisson distribution. This is a case where arrivals to the queuing system occurs at random, but at a certain average rate.

Arrival Populations are eitherLimited (EG: Only people age 21 or over.)Unlimited (EG: cars arriving at a toll booth)Arrival Patterns are eitherRandom (Each arrival is independent)Scheduled (EG: Doctors office visits)Behavior of the ArrivalsBalking (Seeing a long line and avoiding it.)Reneging (Get tired of waiting and leave the line)Jockeying (Switching lines)

The units requiring service enter the queuing system on their arrival and join a queue which is characterized by the maximum permissible number of units that it can contain. A queue is called finite if the number of units in it is finite otherwise it is called infinite.

The service discipline refers to the manner in which the members in the queue are chosen for service.First come, first served (FCFS)Last come, first served (LCFS)Service at random orderService on some priority-procedure

The pattern according to which the customers are served. Here in this unit, we shall deal with queuing models in which service time follows the Exponential and Erlang distributionFacilities given to customers i.e., single channel or multiple channel.

Channels are the paths (ways to get through the system) after getting in line?

EG: McDonalds drive-thru is one channel.

Phases are the number of stops a customer must make, after getting in line?(Single-phase means only one stop for service.) McDonalds drive-thru is a three-phase system: Order Pay Pick-up

Service Facility

Service FacilityService FacilityOnce in line, you have at least two choices of how to get through the system, but only one stop.

Service FacilityService FacilityService FacilityService FacilityOnce in line, you have at least two choices (channels) of how to get through the system and at least two stops (phases).

Service FacilityService FacilityService FacilityService Facility

Service FacilityService FacilityService FacilityService Facility

Queue length. Queue length is defined by the number of persons (customers) waiting in the line at a timeBusy Period. Busy-period of a server is the time during which he remains busy in servicing. Thus it is a time between the start of service of the first unit to the end of service of the last unit in the queue.

Average length of line. Average length of line (or queue) is defined by the number of customers in the queue per unit time.Waiting time. It is the time up to which a unit has to wait in the queue before it is taken into service.Servicing time. The time taken for servicing of a unit is called a servicing time.

Idle Period. When all the units in the queue are served, the idle period of the server begins and it continues up to the time of arrival of the unit (customer). The idle period of a server is the time during which he remains free because there is no customer present in the system.

Mean Servicing Rate. The mean servicing rate for a particular servicing station is defined as the expected number of services completed in a time interval of length unity, given that servicing is going on throughout the entire time unit.Mean Arrival Rate. The mean arrival rate in a waiting line situation is defined as the expected number of arrivals occurring in a time interval of length unity.

Traffic Intensity. In case of a simple queue the traffic intensity is the ratio of mean arrival rate and the mean servicing rate.

Transient StateSteady StateExplosive State

A system is said to be in transient state when its operating characteristics are dependent on time. Thus a queuing system is in transient state when the probability distributions of arrivals , waiting time and servicing time of the customers are dependent on time. This state occurs at the beginning of the operation of the system.

A system is said to be in steady state when its operating characteristics become independent of time. Thus a queuing system acquired steady state when the probability distribution of arrivals, waiting time and servicing time of the customers are independent on time. This state occurs in the long run of the system. In most of the queuing problems, steady state solutions exists independent of the initial state of the queue.

If the arrival rate of the system is more than its servicing rate, the length of the queue will go on increasing with time and will tend to infinity as t tends to infinity. This state of the system is known to be explosive state

1st Attendant2nd AttendantSingle AttendantWhich system is less expensive?(It depends on the relative costs of service versus waiting.)

The costs of waitingLosing customers because of long linesReneging: Customers get tired of waiting and leaveBalking: Customers see a long line and dont get in line.Paying employees to wait for something they need. (waiting for parts, supplies, deliveries, etc.)Unusable (idle) equipment awaiting repairsEG: Broken assembly line machinery.The cost of providing servicePaying people to provide service to customersCustomers can be people, machines, or other objects needing service.

Number of ServersCostsNote that the lowest cost system requires some customer waiting.

Paying repairmen to fix broken machinesPaying dock workers to load and unload trucksPaying customer-service peopleUsing more production people to speed up the lineLeasing of service equipment and facilitiesPaying checkout cashiers

On an average 5 customers reach a barber's shop every hour. Determine the probability that exactly 2 customers will reach in a 30- minute period, assuming that the arrivals follow Poisson distribution?

The manager of a bank observes that, on the average, 18 customers are served by a cashier in an hour. Assuming that the service time has an exponential distribution, what is the probability that (a) a customer shall be free within 3 minutes (b) a customer shall be serviced in more than 12 minutes?

Queue length average number of customers in queue waiting to get serviceSystem lengthaverage number of customers in the systemWaiting time in queue average waiting time of a customer to get serviceTotal time in system average time a customer spends in the system Server idle timerelative frequency with which system is idle

Assumptions:Arrivals are Poisson with a mean arrival rate of, say Service time is exponential, rate being Source population is infiniteCustomer service on first come first served basisSingle service stationFor the system to be workable,

The arrival rate of a customer at a service window of a cinema hall follows a probability distribution with a mean rate of 45 per hour. The service rate of the clerk follows Poisson distribution with a mean of 60 per hour.What is the probability of having customer in the system?What is the probability of having 5 customers in the systemFind average number of customers in the system

The rate of arrival of customers at a public telephone booth follows Poisson distribution with an average time of 10 minutes between one customer and the next. The duration of a phone call is assumed to follow exponential distribution, with mean time of 3 minutes. (i) What is the probability that a person arriving at the booth will have to wait?(ii) Estimate the fraction of a day that the phone will be in use

What is the average length of the non-empty queues that form from time to time? The average number of persons waiting and making telephone calls; andThe MTNL will install a second booth when it is convinced that the customer would expect waiting for at least 3 minutes for their turn to make a call. By how much time should flow of customers increase in order to justify a second booth?

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