Introduction to management science 10e by bernard taylor chapter 12

■ Elements of Waiting Line Analysis ■ The Single-Server Waiting Line System ■ Undefined and Constant Service Times ■ Finite Queue Length ■ Finite Calling Problem ■ The Multiple-Server Wa

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■ Elements of Waiting Line Analysis

■ The Single-Server Waiting Line System

■ Undefined and Constant Service Times

■ Finite Queue Length

■ Finite Calling Problem

■ The Multiple-Server Waiting Line

■ Additional Types of Queuing Systems

Chapter Topics

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 Significant amount of time spent in waiting lines by people, products, etc

 Providing quick service is an important aspect of

quality customer service

 The basis of waiting line analysis is the trade-off

between the cost of improving service and the costs associated with making customers wait

 Queuing analysis is a probabilistic form of analysis

 The results are referred to as operating

characteristics

 Results are used by managers of queuing

operations to make decisions

Overview

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 Waiting lines form because people or

can be served.

 Most operations have sufficient server

run.

 Customers however, do not arrive at a

equal amount of time.

Elements of Waiting Line

Analysis (1 of 2)

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 Waiting lines are continually increasing and decreasing in length and approach an

average service time, in the long run

waiting lines are based on these averages

for customer arrivals and service times.

 They are used in formulas to compute

operating characteristics of the system

which in turn form the basis of decision

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 Components of a waiting line system include

arrivals (customers), servers, (cash

register/operator), customers in line form a

waiting line

 Factors to consider in analysis:

 The queue discipline

 The nature of the calling population

 The arrival rate

 The service rate

The Single-Server Waiting Line System (1 of 2)

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 Queue Discipline: The order in which waiting

customers are served

 Calling Population: The source of customers

(infinite or finite)

 Arrival Rate: The frequency at which customers

arrive at a waiting line according to a probability

distribution (frequently described by a Poisson

distribution)

 Service Rate: The average number of customers

that can be served during a time period (often

described by the negative exponential distribution)

Single-Server Waiting Line System

Component Definitions

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 Assumptions of the basic single-server model:

 An infinite calling population

 A first-come, first-served queue discipline

 Poisson arrival rate

 Exponential service times

 Customers must be served faster than they arrive (

< ) or an infinitely large queue will build up

Single-Server Model

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13-Probability that no customers are in the queuing system:

Probability that n customers are in the system:

Average number of customers in system:

Average number of customer in the waiting line:

Basic Single-Server Queuing

Formulas (1 of 2)

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11

13-Average time customer spends waiting and being served:

Average time customer spends waiting in the

Basic Single-Server Queuing

Formulas (2 of 2)

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Operating Characteristics: Fast Shop

Market (1 of 2)

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13

13-Single-Server Waiting Line System

Operating Characteristics for Fast Shop Market (2 of 2)1

1/[30 -24]

0.167 hour (10 min) avg time in the system per customer

L W

U   



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or waiting line will grow to infinite size.

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15

13-Manager wishes to test several alternatives for

reducing customer waiting time:

1. Addition of another employee to pack up

purchases

2. Addition of another checkout counter

Alternative 1: Addition of an employee

(raises service rate from  = 30 to  = 40

customers per hour)

 Cost $150 per week, avoids loss of $75 per

week for each minute of reduced customer waiting time

 System operating characteristics with new

Effect of Operating Characteristics

(1 of 6)

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(2 of 6)

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13-Alternative 2: Addition of a new checkout counter ($6,000

plus $200 per week for additional cashier).

  = 24/2 = 12 customers per hour per checkout

counter

  = 30 customers per hour at each counter

 System operating characteristics with new

parameters:

Po = 60 probability of no customers in the system

L = 0.67 customer in the queuing system

Lq = 0.27 customer in the waiting line

W = 0.055 hour per customer in the system

Wq = 0.022 hour per customer in the waiting line

U = 40 probability that a customer must wait

I = 60 probability that server is idle

(3 of 6)

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13-Savings from reduced waiting time worth:

$500 per week - $200 = $300 net savings per week.

After $6,000 recovered, alternative 2 would

provide:

$300 -281.25 = $18.75 more savings per week

(4 of 6)

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13-Table 13.1 Operating Characteristics for Each Alternative

System

(5 of 6)

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13-Figure 13.2 Cost Trade-Offs for Service Levels

(6 of 6)

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13-Exhibit 13.1

Solution with Excel and Excel QM (1

of 2)

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13-Exhibit 13.2

of 2)

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13-Exhibit 13.3

Solution with QM for Windows

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13- Constant, rather than exponentially

distributed service times, occur with

machinery and automated equipment.

 Constant service times are a special case of

the single-server model with undefined service times.

L W

Undefined and Constant Service

Times

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25

13-0 1 1 20 33 probability that machine not in use

30

2 2 / 20 1/15 20/ 30

2 1 / 2 1 20/30 3.33 employees waiting in line

3.33 (20/ 30) 4.0 employees

 in line and using the machine

 Data: Single fax machine; arrival rate of 20

users per hour, Poisson distributed; undefined service time with mean of 2 minutes,

standard deviation of 4 minutes.

 Operating characteristics:

Undefined Service Times Example (1

of 2)

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q q

q

L W

 Operating characteristics (continued):

Undefined Service Times Example (2

of 2)

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13- In the constant service time model there is no variability in service times;  = 0.

 Substituting  = 0 into equations:

 All remaining formulas are the same as the

Constant Service Times Formulas

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L W

Constant Service Times Example

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13-Exhibit 13.4

Times

Solution with Excel

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13-Exhibit 13.5

Times

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(1 )

n

M q

q M

 Operating characteristics, where M is the

maximum number in the system:

Finite Queue Length

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13-Metro Quick Lube single bay service; space for one vehicle in service and three waiting for service;

mean time between arrivals of customers is 3

minutes; mean service time is 2 minutes; both

inter-arrival times and service times are

exponentially distributed; maximum number of

vehicles in the system equals 4

Operating characteristics for  = 20,  = 30, M = 4:

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13-Average queue lengths and waiting times:

1 ( 1)( / )

/

1 / 1 ( / ) 1

5 (5)(20/ 30) 20/ 30 1.24 cars in the system

5

1 20/30 1 (20/30)

(1 ) 1.24 20(1 076) 0.62 cars waiting

30 1.24 0.067 hours waiting in the s (1 ) 20(1 076)

M q

M

M M

L

M L

P

L L

L W

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13-Exhibit 13.6

Finite Queue Model Example

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13-Exhibit 13.7

Finite Queue Model Example

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0 where N population size, and n 1, 2, N

! (1 ) ( )!

1 (1 )

n N

 Operating characteristics for system with

Poisson arrival and exponential service times:

Finite Calling Population

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13-Wheelco Manufacturing Company; 20

machines; each machine operates an average

of 200 hours before breaking down; average

time to repair is 3.6 hours; breakdown rate is Poisson distributed, service time is

exponentially distributed.

Is repair staff sufficient?

 = 1/200 hour = 005 per hour

 = 1/3.6 hour = 2778 per hour

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13-…System seems woefully inadequate.

20 20! .005 (20 )! 2778 0

.005 2778

20 1 652 169 machines waiting

.005 169 (1 652) 520 machines in the system

.169 1.74 hours waiting for repair (20 520)(.005)

1 1.74

q

n n

L L W W

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13-Exhibit 13.8

Finite Calling Population Example

of 2)

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13-Exhibit 13.9

of 2)

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13-Exhibit 13.10

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13-Multiple-Server Waiting Line (1 of 3)

Figure 13.3

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13- In multiple-server models, two or more

independent servers in parallel serve a

single waiting line.

 Biggs Department Store service

department; first-come, first-served basis.

Multiple-Server Waiting Line (2

of 3)

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13-Multiple-Server Waiting Line

Queuing Formulas (1 of 3)

 Assumptions:

 First-come first-served queue discipline

 Poisson arrivals, exponential service times

 Infinite calling population

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13-0

average number of customers in the queue

1 average time customer is in the queue

1 probability customer must wait for service

!

q

q q

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Multiple-Server Waiting Line

Biggs Department Store Example (1

of 2) = 10,  = 4, c = 3

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13-10 6

4 3.5 customers on the average waiting to be served

3.5 10 0.35 hour average waiting time in line per customer

3 3(4)

1 10 (.045) 3! 4 3(4) 10

703 probability customer must wait for servi

Biggs Department Store Example (2

of 2)

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13-Exhibit 13.11

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13-Exhibit 13.12

Solution with Excel QM

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13-Exhibit 13.13

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13-Figure 13.4 Single Queues with Single and Multiple

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13-Other items contributing to queuing systems:

 Systems in which customers balk from

entering system, or leave the line (renege ).

 Servers who provide service in other than first-come, first-served manner

 Service times that are not exponentially

distributed or are undefined or constant

 Arrival rates that are not Poisson

distributed

 Jockeying (i.e., moving between queues)

Additional Types of Queuing

Systems (2 of 2)

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55

13-Problem Statement: Citizens Northern

Savings Bank loan officer customer interviews Customer arrival rate of four per hour, Poisson distributed; officer interview service time of 12 minutes per customer.

1 Determine operating characteristics for

this system.

2 Additional officer creating a

multiple-server queuing system with two channels Determine operating characteristics for this system.

Example Problem Solution (1 of

5)

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L q =  2 / ( - ) = 4 2 / 5(5 - 4) = 3.2 customers on average in the

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13-Step 1 (continued):

W = 1 / ( - ) = 1 / (5 - 4) = 1 hour on average in the system

W q =  / (u - ) = 4 / 5(5 - 4) = 0.80 hour (48 minutes) average time in the waiting line

P w =  /  = 4 / 5 = 80 probability the new accounts officer is busy and a

customer must wait

Example Problem Solution (3 of 5)

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( / )

2 ( 1)!( ) 0.952 average number of customers in the system

Step 2: Determine the Operating

Characteristics for the Multiple-Server

System.

 = 4 customers per hour arrive;  = 5

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13-0.152 average number of customers in the queue

1 0.038 hour average time customer is in the queue

1

! 229 probability customer must wait for service

q

q q

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