WAITING LINES and QUEUING THEORY MODELS

Isaac Lawrence Yvan C. Andaya

INTRODUCTION

• The Foundation of modern queuing theory is based on studies about automatic dialing equipment made in the early part of the 20th century by Danish telephone engineering “A.K. ERLANG”.

QUEUING THEORY

• The Planning and Analysis of service capacity.• It is the study of Waiting Lines.• Three basic components of a queuing process are

arrivals, service facilities and the actual waiting line.

• Queuing theory has been used for operations research, manufacturing and system analysis.

Applications of Queuing Theory

• Telecommunications• Traffic Control• Determining the sequence of computer

operations• Airport traffic, airline ticket sales• Predicting computer performance

• Waiting lines tends to form even though a system is basically under loaded. For example: a fast food restaurant may have the capacity to handle an average of 100 orders/hour but may experience waiting lines even though the average number of orders is only 75/hour. The operative word is “AVERAGE”.

• In real situation, customers arrive at random intervals rather than at evenly spaced intervals, also some orders take longer to fill than the others. In short both arrivals and service time exhibit a high degree of variability, w/c results to temporary overloading – waiting lines.

Implication of Waiting Lines

There are Reasons why managers have to be concerned with waiting lines:1.The cost of providing a waiting space.2.Should customers leave the line before being served or refuse to wait at all, the possible business loss should be considered.3.A possible loss of goodwill.4.Probable reduction in customer satisfaction.5.The resulting congestion may interrupt other business operations and or customers.

WAITING LINE COSTS

• The Main goal of waiting line analysis is to minimize costs. There are two type of cost in a queuing scenario; those costs relating to customers waiting for service and those relating to capacity.

Capacity costs

• Is defined as the costs of maintaining the ability to provide service. Capacity is lost when a service facility is idle, since capacity cannot be stored.

• Meanwhile the cost of customer waiting for service could include: the salaries paid to employees while they wait, the cost of space for waiting and any loss of business due to customers refusing to wait and or going elsewhere. This loss of business is oftentimes called as ”Opportunity Loss.”

• Queuing costs and service level

*Optimal Service Level

Cos

t

Service Level

Cost of Providing Service

Total Expected Cost

Cost of Waiting Time

Characteristics of a Queuing System

• There are three parts to a queuing system1. The arrivals or inputs to the system (sometimes

referred to as the calling populationcalling population)2. The queue or waiting line itself3. The service facility

• These components have their own characteristics that must be examined before mathematical models can be developed

Arrival Characteristics

• Arrival Characteristics have three major characteristics, sizesize, patternpattern, and behaviorbehavior– Size of the calling population

• Can be either unlimited (essentially infiniteinfinite) or limited (finitefinite)

– Pattern of arrivals• Can arrive according to a known pattern or can

arrive randomlyrandomly• Random arrivals generally follow a Poisson Poisson

distributiondistribution

• The Poisson distribution is

4,... 3, 2, 1, 0, for

XXeXP

X

!)(

P(X) = probability of X arrivalsX = number of arrivals per unit of time = average arrival ratee = 2.7183

POISSON DISTRIBUTION

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X

%.)(.!

)( 14135301

113530020

02

eP

%.)(.!

)( 27270601

2135301

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eeP

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222

222

eeP

POISSON DISTRIBUTION

Behavior of arrivals

– Join the Queue, Join the Queue, Most queuing models assume customers are patient and will wait in the queue until they are served and do not switch lines

– BalkingBalking refers to customers who refuse to join the queue

– RenegingReneging customers enter the queue but become impatient and leave without receiving their service

– That these behaviors exist is a strong argument for the use of queuing theory to managing waiting lines

– Waiting lines can be either limitedlimited or unlimitedunlimited– Queue discipline refers to the rule by which

customers in the line receive service– The most common rule is first-in, first-outfirst-in, first-out (FIFOFIFO)– Other rules are possible and may be based on

other important characteristics– Other rules can be applied to select which

customers enter which queue, but may apply FIFO once they are in the queue

Waiting Line Characteristics

Service Facility Characteristics

• Service systems are classified in terms of the number of channels, or servers, and the number of phases, or service stops

• A single-channel systemsingle-channel system with one server is quite common

• MultichannelMultichannel systemssystems exist when multiple servers are fed by one common waiting line

• In a single-phase systemsingle-phase system the customer receives service form just one server

• If a customer has to go through more than one server, it is a multiphase systemmultiphase system

Service Time Distribution

– Service patterns can be either constant or random– Constant service times are often machine controlled– More often, service times are randomly distributed

according to a negative exponential probability negative exponential probability distributiondistribution

– Models are based on the assumption of particular probability distributions

– Analysts should take to ensure observations fit the assumed distributions when applying these models

Performance Measure of Queuing systems

Managers observed these five factors when evaluating

existing or proposed service systems.•The Average number of customers waiting, either in line or in the system.•The Average time customers wait or spend in the cue, either in line or in the system.•Average length of the cue.•System Utilization – refers to the percentage of capacity utilized.•The implied cost of a given level of capacity and its related waiting line.•The probability that an arrival will have to wait for service.•Probability that the service will be idle.

SINGLE CHANNEL EXPONENTIAL SERVICE TIME MODEL

• This model involves a system that has one

server and there is no limit on length of queue.1.Queue discipline: FIFO2.No Balking or Regening3. Independent arrivals4.Arrivals: Poisson distributed5.Service times: negative exponential6.Average service rate . Average arrival rate