Queuing Theory

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Queuing Theory

• Chapter 12

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12.1 Introduction

• Queuing is the study of waiting lines, or queues.
• The objective of queuing analysis is to design
  systems that enable organizations to perform
  optimally according to some criterion.
• Possible Criteria
• Maximum Profits.
• Desired Service Level.

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• Analyzing queuing systems requires a clear
  understanding of the appropriate service
  measurement.
• Possible service measurements
• Average time a customer spends in line.
• Average length of the waiting line.
• The probability that an arriving customer must
  wait for service.

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• The Arrival Process
• There are two possible types of arrival processes
• Deterministic arrival process.
• Random arrival process.
• The random process is more common in businesses.
• Under three conditions a Poisson distribution can
  describe the random arrival process.

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• The Poisson Arrival Distribution

Where l mean arrival rate per time
unit. t the length of the interval. e
2.7182818 (the base of the natural
logarithm). k! k (k -1) (k -2) (k -3) (3)
(2) (1).

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• The Waiting Line
• Line configuration
• A Single service Queue.
• Multiple service queue with single waiting line.
• Multiple service queue with multiple waiting
  lines.
• Tandem queue (multistage service system).
• Jockeying
• Jockeying occurs if customers switch lines when
  they perceived that another line is moving
  faster.
• Balking
• Balking occurs if customers avoid joining the
  line when they perceive the line to be too long.

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• Priority rules
• Priority rules define the line discipline.
• These rules select the next customer for service.
• There are several commonly used rules
• First come first served (FCFS).
• Last come first served (LCFS).
• Estimated service time.
• Random selection of customers for service.
• Homogeneity
• An homogeneous customer population is one in
  which customers require essentially the same type
  of service.
• A Nonhomogeneous customer population is one in
  which customers can be categorized according to
• Different arrival patterns
• Different service treatments.

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• The Service Process
• Some service systems require a fixed service
  time.
• In most business situations, however, service
  time varies widely among customers.
• When service time varies, it is treated as a
  random variable.
• The exponential probability distribution is used
  sometimes to model customer service time.

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• The Exponential Service Time Distribution

where m is the average number of customers
who can be served per time
period.
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Schematic illustration of the exponential
distribution
f(X)
The probability that service is completed within
t time units
X t
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12.3 Measures of Queuing System Performance

• Performance can be measured by focusing on
• Customers in queue.
• Customers in the system.
• Transient and steady state periods complicate the
  service time analysis

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• The transient period occurs at the initial time
  of operation.
• Initial transient behavior is not indicative of
  long run performance.
• The steady state period follows the transient
  period.
• In steady state, long run probabilities of having
  n customers in the system do not change as time
  goes on.
• In order to achieve steady state, the effective
  arrival rate must be less than the sum of
  effective service rates .
• llt m llt m1 m2mk
  llt km
• For one server For k servers For k servers
  each with service rate m

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• Steady State Performance Measures
• P0 Probability that there are no customers in
  the system.
• Pn Probability that there are n customers in
  the system.
• L Average number of customers in the system.
• Lq Average number of customers in the queue.
• W Average time a customer spends in the
  system.
• Wq Average time a customer spends in the queue.
• Pw Probability that an arriving customer must
  wait for service.
• r Utilization rate for each server (the
  percentage of time that each server is
  busy).

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• Littles Formulas
• Littles Formulas represent important
  relationships between L, Lq, W, and Wq.
• These formulas apply to systems that meet the
  following conditions
• Single queue systems,
• Customers arrive at a finite arrival rate l,
  and
• The system operates under steady state
  condition.
• L l W Lq l Wq L Lq l /
  m
• For the infinite population case

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• Classification of Queues
• Queuing system can be classified by
• Arrival process.
• Service process.
• Number of servers.
• System size (infinite/finite waiting line).
• Population size.
• Notation
• M (Markovian) Poisson arrivals or exponential
  service time.
• D (Deterministic) Constant arrival rate or
  service time.
• G (General) General probability for arrivals or
  service time.

Example M / M / 6 / 10 / 20
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12.4 M / M / 1 Queuing System

• Characteristics
• Poisson arrival process.
• Exponential service time distribution.
• A single server.
• Potentially infinite queue.
• An infinite population.

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• Performance Measures for the M / M /1 Queue
• P0 1- (l / m)
• Pn 1 - (l / m) (l/ m)n
• L l / (m - l)
• Lq l 2 / m(m - l)
• W 1 / (m - l)
• Wq l / m(m - l)
• Pw l / m
• r l / m

The probability that a customer waits in the
system more than t is P(Xgtt) e-(m - l)t
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MARYs SHOES

• Customers arrive at Marys Shoes every 12 minutes
  on the average, according to a Poisson process.
• Service time is exponentially distributed with an
  average of 8 minutes per customer.
• Management is interested in determining the
  performance measures for this service system.

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SOLUTION

• Input
• l 1/ 12 customers per minute 60/ 12 5 per
  hour.
• m 1/ 8 customers per minute 60/ 8 7.5
  per hour.
• Performance Calculations

P0 1- (l / m) 1 - (5 / 7.5) 0.3333 Pn 1
- (l / m) (l/ m) (0.3333)(0.6667)n
L l / (m
- l) 2 Lq l2/ m(m - l) 1.3333 W 1 /
(m - l) 0.4 hours 24 minutes Wq l / m(m -
l) 0.26667 hours 16 minutes
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12.5 M / M / k Queuing Systems

• Characteristics
• Customers arrive according to a Poisson process
  at a mean rate l.
• Service time follow an exponential distribution.
• There are k servers, each of which works at a
  rate of m customers.
• Infinite population, and possibly infinite line.

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• Performance measure

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The performance measurements L, Lq, Wq,, can be
obtained from Littles formulas.
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LITTLE TOWN POST OFFICE

• Little Town post office is opened on Saturdays
  between 900 a.m. and 100 p.m.
• Data
• On the average 100 customers per hour visit the
  office during that period. Three clerks are on
  duty.
• Each service takes 1.5 minutes on the average.
• Poisson and Exponential distribution describe the
  arrival and the service processes respectively.

• The Postmaster needs to know the relevant service
  measures in order to
• Evaluate current service level.
• Study the effects of reducing the staff by one
  clerk.

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SOLUTION

• This is an M / M / 3 queuing system.
• Input
• l 100 customers per hour.
• m 40 customers per hour (60 / 1.5).
• Does steady state exist (l lt km )?
• l 100 lt km 3(40) 120.

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12.6 M / G / 1 Queuing System

• Assumptions
• Customers arrive according to a Poisson process
  with a mean rate l.
• Service time has a general distribution with mean
  rate m.
• One server.
• Infinite population, and possibly infinite line.

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• Pollaczek - Khintchine Formula for L

Note It is not necessary to know the particular
service time distribution. Only the mean and
standard deviation of the distribution are needed.
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TEDS TV REPAIR SHOP

• Teds repairs television sets and VCRs.
• Data
• It takes an average of 2.25 hours to repair a
  set.
• Standard deviation of of the repair time is 45
  minutes.
• Customers arrive at the shop once every 2.5 hours
  on the average, according to a Poisson process.
• Ted works 9 hours a day, and has no help.
• He considers purchasing a new piece of equipment.
• New average repair time is expected to be 2
  hours.
• New standard deviation is expected to be 40
  minutes.

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Ted wants to know the effects of using the new
equipment on - 1. The average number of sets
waiting for repair 2. The average time a
customer has to wait to get his repaired set.
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SOLUTION

• This is an M / G / 1 system (service time is not
  exponential (because s 1/m).
• Input
• The current system (without the new equipment)
• l 1/ 2.5 0.4 customers per hour.
• m 1/ 2.25 0.4444 customers per hour.
• s 45/ 60 0.75 hours.
• The new system (with the new equipment)
• m 1/2 0.5 customers per hour.
• s 40/ 60 0.6667 hours.

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12.7 M / M / k / F Queuing System

• Many times queuing systems have designs that
  limit their size.
• When the potential queue is large, an infinite
  queue model gives accurate results, even though
  the queue might be limited.
• When the potential queue is small, the limited
  line must be accounted for in the model.

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• Characteristics of the M / M / k / F system
• Poisson arrival process at mean rate l.
• k servers, each having an exponential service
  time with mean rate m.
• Maximum number of customers that can be present
  in the system at any one time is F.
• Customers are blocked (and never return) if the
  system is full.

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• The Effective Arrival Rate
• A customer is blocked if the system is full.
• The probability that the system is full is PF.
• The effective arrival rate the rate of arrivals
  that make it through into the system (le).

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RYAN ROOFING COMPANY

• Ryan gets most of its business from customers who
  call and order service.
• Data
• One appointment secretary takes phone calls from
  3 telephone lines.
• Each phone call takes three minutes on the
  average.
• Ten customers per hour call the company on the
  average.

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• When a telephone line is available but the
  secretary is busy serving a customer, a new
  calling customer is willing to wait until the
  secretary becomes available.
• When all the lines are busy, a new calling
  customer gets a busy signal and calls a
  competitor.
• Arrival process is Poisson, and service process
  is Exponential.

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• Management would like to design the following
  system
• The fewest lines necessary.
• At most 2 of all callers get a busy signal.
• Management is interested in the following
  information
• The percentage of time the secretary is busy.
• The average number of customers kept on hold.
• The average time a customer is kept on hold.
• The actual percentage of callers who encounter
  a busy signal.

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SOLUTION

• This is an M / M / 1 / 3 system
• Input
• l 10 per hour.
• m 20 per hour (1/ 3 per minute).
• WINQSB gives
• P0 0.5333, P1 0.2667 , P2 0. 1333 , P3
  0.0667
• 6.7 of the customers get a busy signal.
• This is above the goal of 2.

M / M / 1 / 4 system
M / M / 1 / 5 system
P0 0.516, P1 0.258, P2 0.129, P3 0.065,
P4 0.032 3.2 of the customers get the
busy signal Still above the goal of 2
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12.8 M / M / 1 / / m Queuing Systems

• In this system the number of potential customers
  is finite and relatively small.
• As a result, the number of customers already in
  the system effects the rate of arrivals of the
  remaining customers.
• Characteristics
• A single server.
• Exponential service and interarrivall time,
  Poisson arrival process.
• A population size of m customers (m is finite).

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PACESETTER HOMES

• Pacesetter Homes runs four different development
  projects.
• Data
• A stoppage occurs once every 20 working days on
  the average in each site.
• It takes 2 days on the average to solve a
  problem.
• Each problem is handled by the V.P. for
  construction.
• How long on the average a site does not operate?
• With 2 days to solve a problem (current
  situation)
• With 1.875 days to solve a problem (new situation)

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SOLUTION

• This is an M / M / 1 // 4 system.
• The four sites are the four customers.
• The V.P. for construction can be considered a
  server.
• Input
• l 0.05 (1/ 20)
• m 0.5 (1/ 2 using the current car).
• m 0.533 (1/1.875 using a new car).

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Results obtained from WINQSB
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12.9 ECONOMIC ANALYSIS OF QUEUING SYSTEMS

• The performance measures previously developed are
  used next to determine a minimal cost queuing
  system.
• The procedure requires estimated costs such as
• Hourly cost per server .
• Customer goodwill cost while waiting in line.
• Customer goodwill cost while being served.

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WILSON FOODS TALKING TURKEY HOT LINE

• Wilson Foods has an 800 number to answer
  customers questions.
• Data
• On the average 225 calls per hour are received.
• An average phone call takes 1.5 minutes.
• A customer will stay on the line waiting at most
  3 minutes.
• A customer service representative is paid 16 per
  hour.
• Wilson pays the telephone company 0.18 per
  minute when the customer is on hold or when being
  served.
• Customer goodwill cost is 20 per minute while on
  hold.
• Customer goodwill cost while in service is 0.05.

How many customer service representatives
should be used to minimize the hourly cost of
operation?
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SOLUTION

• The total cost model

Average hourly goodwill cost for customers on
hold
Total hourly wages
Total average hourly Telephone charge
Average hourly goodwill cost for customers in
service
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• Input
• Cw 16
• Ct 10.80 per hour 0.18(60)
• gw 12 per hour 0.20(60)
• gs 0.05 per hour 0.05(60)
• The Total Average Hourly Cost is
• TC(K) 16K (10.83)L (12 - 3)Lq
• 16K 13.8L 9Lq

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• Assuming a Poisson arrival process and an
  Exponential service time, we have an M / M / K
  system.
• l 225 calls per hour.
• m 40 per hour (60/ 1.5).
• The minimal possible value for K is 6 to ensure
  that steady state exists (lltKm).
• WINQSB was used to generate results for L, Lq,
  and Wq.

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• Summary of results of the runs for k6,7,8,9,10

Conclusion employ 8 customer service
representatives.
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Example M/M/1 vs M/M/2

• Which is better?
• Select 1 machine with certain speed and certain
  cost or 2 machines with half speed and half
  cost?
• M/M/1
• Input
• l 20 m 30 Cw 5 gw 2 gs
  1
• M/M/2
• Input
• l 20 m 15 Cw 2.5 gw 2 gs
  1

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Example M/M/1 vs M/M/2 (cont)
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Example M/M/1 vs M/M/2 (cont)
Difference 8.4667 - 8.3333 0.1334 /hour
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12.10 Tandem Queuing Systems

• In a Tandem Queuing System a customer must visit
  several different servers before service is
  completed.
• For cases in which customers arrive according to
  a Poisson process and service time in each
  station is Exponential,

Total Average Time in the System Sum of all
Average Times in the Individual Stations
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BIG BOYS SOUND, INC.

• Big Boys sells audio merchandise.
• The sale process is as follows
• A customer places an order with a sales person.
• The customer goes to the cashier station to pay
  for the order.
• After paying, the customer is sent to the pickup
  desk to obtain the good.

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• Data for a regular Saturday
• Personnel.
• 8 sales persons are on the job.
• 3 cashiers.
• 2 workers in the merchandise pickup area.
• Average service times.
• Average time a sales person wait on a customer is
  10 minutes.
• Average time required for the payment process is
  3 minutes.
• Average time in the pickup area is 2 minutes.
• Distributions.
• Exponential service time in all the service
  stations.
• Poisson arrival with a rate of 40 customers an
  hour.

Only 75 of the arriving customers make a
purchase.
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SOLUTION

• This is a Three Station Tandem Queuing System

M / M / 2
M / M / 3
l 30
M / M / 8
l 30
l 40
2.67 minutes
W2 3.47 minutes
Total 20.14 minutes.
W1 14 minutes