Unit 4 queuing models

Queuing Models

Unit 4

2Contents

• Characteristics of queuing systems• Queuing notation• Simulation Examples:

• Queuing• Inventory System

3Introduction

• Simulation is often used in the analysis of queuing models• Queueing models whether solved mathematically or analysed

through simulation, provide the analyst a powerful tool for designing and evaluating the performance of queuing systems.• Measures of system performance include

• Server utilization (percentage of time a server is busy)• Length of waiting lines• Delays of customers

• Simple queueing model is shown below

4Simple queueing model

5Simple queueing model

• Customers arrive from time to time and join a queue (waiting line)• They are eventually served and then finally they leave the

system• Customers refers to any type of entity that can be viewed

as requesting a service from a system• Eg. : service facilities, production systems, repair and

maintenance facilities, communications and computer systems and transport and material handling systems

6Characteristics of queuing system

• The key elements of queuing system are the customers and server• Customer refers to anything that arrives at a facility and

requires service • Eg. people, machines, truck, mechanics, patients, pallets, airplanes ,

email, cases , orders , or dirty clothes• Server refers to anything that provides the requested service.

• Eg. receptionists, repair personnel, mechanics, medical personnel, automatic storage and retrival machines such as cranes, runways at airport, automatic packers, order pickers, washing machines, CPU in computers

7Examples of queueing system

8Characteristics of queuing system

• Calling population

• System capacity

• The Arrival Process

• Queue Behavior and Queue Discipline

• Service Times and the service Mechanism

9The calling population

• The population of potential customers, is referred as the calling population which may be finite or infinite• Consider the following scenario to understand the terms calling

population, customers and server• Consider the personal computers of the employees of a small

company that are supported by the IT staff of three technicians• When a computer fails, needs new software etc, it is attended

by one of IT staff.• Computers are the customers, IT staff is a server and calling

population is finite here, consists of the personal computers at the company.

10The calling population

• The systems with large population of potential customers, the calling population is assumed to be infinite.• The difference between the finite and infinite calling

population is how the arrival rate is defined.• In infinite calling population, the arrival rate is not affected by

the number of customers left the calling population and joined the queuing system• In finite calling population, the arrival rate to the queueing

system does depend on the number of customers being served and waiting.

11The calling population

• The arrival rate defined as the expected number of arrivals in the next unit of time• Eg. Consider a hospital with 5 patients assigned to a single

nurse. • When all the patients are resting , the nurse is idle hence

the arrival rate is maximum since any of the patients can call nurse for assistance next instant• When all the 5 patients have called the nurse then arrival

rate is zero i.e. no arrival is possible until the nurse finishes with a patient.

12System Capacity

• The limit to the number of customers that may be in the waiting line or system• Eg. Automatic car wash might have room for 10 cars to

wait in a line to enter into the mechanism• When the system capacity is reached, the new customers

immediately joins the calling population

13System capacity

• When the system is with limited capacity, distinction is made between arrival rate and effective arrival rate• Arrival rate number of arrivals per time unit• Effective arrival rate the number who arrive and enter the

system per unit time

14The arrival process

• The arrival process for infinite population models is usually characterized in terms of interarrival times of successive customers.• May occur in scheduled times or random times• Customers may arrive one at a time or in batches.• The bacths may be of constant size or of random size.• Most important model for random arrivals is Poisson arrival

process.

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• If An represents the interarrival time in between customer n-1 and customer n then for Poisson arrival process An is exponentially distributed with the mean 1/λ time units.• The arrival rate is λ customers per unit time. • Eg. Arrival of people for resturants, banks, arrival of

telephone calls at call center, the arrival of demands, orders for a service or product arrival of failed components machines for a repair facility.,

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• Second type of arrivals is scheduled arrivals, such as patients to a doctor’s office or scheduled airline flight arrivals to an airport• Third type of arrival is when at least a customer is assumed

to always to be present in the queue so that the server is never idle because of lack of customers.• In case of finite population models arrival process is

classified as pending and not pending• Customer is defined as pending when customer is outside

the queuing system and a member of calling population

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• Customer is defined as not pending when the customer gets served by the server • Eg. In a hospital the patients are pending when they are

resting and becomes not pending the instant they call for the nurse• Runtime is defined for every customer i.e. length of time

from departure from the queuing system until the next customer arrives into the queue.

18Arrival process for a finite population model

19Queuing behaviour and queuing discipline• Queue behaviour refers to the actions of the customers while

in a queue waiting for a service to begin• Incoming customers will• Balk – leave when they see that the line is too

long• Renege- leave after being in the line when they

see the line is moving to slow• Jockey- move from one line another if they think

they have chosen a slow line

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• Queue discipline refers to logical ordering of customers in a queue and determines which customers will be chosen for service when a server becomes free• Some queue disciplines include FIFO, LIFO, service in

random order (SIRO), shortest processing time first(SPT), service according to priority (PR)• In FIFO, the service begin in the same order as arrivals but

the customers can leave the system because of different length service times

21Service times and service mechanism

• Service times of successive arrivals are denoted by S1,S2,S3 … • They may be constant or random.• Customers can have same service times for a class or type

of customers• Some times, different customers can also have different

service time distributions• Service time may depend on time of day or upon the length

of waiting line

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• Queueing system consists of number of service centers and interconnecting queues.• Parallel service mechanisms are either single server,

multiple server or unlimited server• Self service facility is usually characterized as having

unlimited number of servers.

23Eg. Warehouse

• Customers may either serve themselves or wait for one of three clerks and then finally leave after paying at a single cashier.• The system flow is shown in the following figure• The subsystem, consisting of queue 2 and service center 2

is shown in the figure

24Discount warehouse with three service centers

25Service center 2, with c = 3 parallel servers

26Ex. Candy manufacturer

• Has a production line that consists of three machines separated by inventory in process buffers• First machine makes and wraps the individual pieces of

candy• Second packs 50 pieces in a box• Third machine seals and wraps the box.• The inventory buffers have a capacity of 1000 boxes each• Machine 1 shuts down whenever its inventory buffer fills to

capacity and machine 2 shuts down whenever its buffer empties.

27Candy production line

28Queuing notation

• Recognizing the diversity of queuing systems, Kendall proposed a national system for parallel server systems which has been widely adopted.• The model is based on the format A/B/c/N/K. these letters

represent the following system characteristics:A the inter arrival time distributionB the service time distributionsC the number of parallel serversN the system capacityK the size of the calling population

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Common symbols for A & B include • M (exponential or Markov)• D ( Constant or deterministic)• Ek (Erlang of order k)• PH (phase-type)• H ( hyper exponential)• G ( arbitrary or general)• G1 ( general independent)

• Eg: M/M/1/∞/∞ indicates a single- server system that has unlimited queue capacity and an infinite population

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31Long-Run measures of performance of queuing systems

• Time Average Number in system ( L )• The number of customers in a queue (LQ)• Average Time Spent in System per Customer ( w )• The conservation Equation: L = λw• Server Utilization• Costs in queuing problems

32Time average number in system (L)

33Time Average Number in System L

• Consider a queuing system over a period of Time T, & let L(t) denote the no. of customers in the system at time t. let Ti denote the total time during [0,T] in which the system contained exactly i customers. The time –weighted-average number in a system is defined by•

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• Many queuing systems exhibit a certain kind of long-run stability in terms of their average performance . for such condition, the long run time-average number in system , with probability 1 can be given as

35The number of customers in a queue (LQ)

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37 Average Time Spent in System Per Customer w

• If the simulation is done for a period of time , say T, then record the time each customer spend in the system during [0,T], say W1,W2…… WN where N is the number of arrivals during [0,T]. The average time spent in system per customer, called average system time given as, where w is called long run average system time.

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• If the system under consideration is the queue alone, then the equation can be written as

39example

• For the system history shown in figure 1 N=5 customers. The system has a single server and FIFO queue discipline. • Arrivals occurs at the rate of 0,3,5,7 and 16.• Departure time 2,8,10,14,20• Find the average time spent in system per customer

40figure

41The conservation Equation : L = λW

• Consider a system with N= 5 arrivals in T=20 time units and thus the observed arrival rate was λ = N/T.• The relationship between L,λ, W is not coincidental. • It holds for almost all queuing systems or subsystems

regardless of the number of servers, the queue discipline, or any other special circumstances allowing T∞ and N ∞ equation becomes L = λw, where λ is the long-run average arrival rate and the equation is called conservation equation.

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• It says that “ the average number of customers in the system at an arbitrary point in time is equal to the product of average number of arrivals per time unit, times the average time spent in the system.• The total system time of all customers is given by the total

area under the number-in-system function L(t)

43Server Utilization

• Defined as the proportion of time that a server is busy. Long –run server utilization is denoted by ρ. For the systems that exhibit long run stability

51M/M/1 queue

• M/M/1 queue will often be a useful approximate model

when service times have standard deviations approximately equal to their means.

• The different steady state parameters can be calculated by substituting σ² = 1 / μ² in the steady state parameter values of M/G/1 queueing model

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66problem

• The inter arrival times and service times at a single –chair saloon have been shown to exponentially distributed. The values of λ and μ are 2 per hour and 3 per hour respectively. For this M/M/1 queue determine

1. the time average number of customers in the system2. The average time an arrival spends in system3. The average time the customer spends in the queue4. The time average number in the queue.

67Question bank

• Explain the characteristics of queuing system• Explain the queueing notation A/B/C/N/K with an example• Explain the steady state parameters of M/M/1 queue• Explain Long-Run measures of performance of queuing

systems• Give all the queueing notation of parallel server system• Write short note on network of queues• Dump truck and inventory system problems

68Unit 5 questions

• Explain a single server queue simulation in java• Explain the event schedule algorithm and list processing

operation• Write the GPSS block diagram for single server queue

simulation• What is boot strapping ? Explain time advance algorithm

End of unit 4Thank you