Queuing Theory

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Queuing Theory

• Queuing Theory represents the body of knowledge dealing with waiting lines.

• Most queuing problems focus on determining the level of service that a company should provide.

Service Level

$

Total Cost

Cost of Providing Service

Cost of Customer Dissatisfaction

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Queuing Theory

Server

CustomerArrives

Waiting Line

Server 1

Server 2

Server 3

Waiting Line

CustomerArrives

CustomerLeaves

CustomerLeaves

CustomerLeaves

CustomerLeaves

CustomerArrives

CustomerArrives

CustomerArrives

CustomerLeaves

CustomerLeaves

CustomerLeaves

Waiting Line

Waiting Line

Waiting Line

Server 1

Server 2

Server 3

Queuing Systems Configurations

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Queuing Theory

• Generation of Customers– Infinite vs. Finite calling

population– Homogeneity of the calling

population– Individual vs. Batch arrivals– Deterministic vs. Stochastic

arrivals• Queuing of Customers

– Single vs. Multiple servers– Finite vs. Infinite queues

Characteristics of a Queuing Process

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Queuing Theory

– FIFO vs. LIFO disciplines– Priority rules

• Servicing the Customers– Deterministic vs. Stochastic

service time– Individual vs. Batch Processing


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Queuing Theory

• Generation of Customers – Poisson probability distribution

‘x’ represents the number of arrivals in a specific time period.

‘’ represents the ‘arrival rate’, that is, the average number of arrivals per time period.


p x ex

x

( )!

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Queuing Theory

The time between arrivals is known as the interarrival time. If the number of arrivals in a given period follows a Poisson distribution, with mean , the interarrival times follow an exponential probability distribution with mean 1/

The exponential distribution exhibit the memoryless property. An arrival process is memoryless if the time until the next arrival occurs does not depend on how much time has elapsed since the last arrival.

Arrival Rate

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Queuing TheoryArrival Rate

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Queuing Theory

Queue time is the amount of time a customer spends waiting in line for service to begin.

Service time is the amount of time a customer spends at a service facility once the actual performance of service begins.

Service time is often model as an exponential random variable

Service Rate

p t T e for T 1 0 T ,

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Queuing Theory

The service rate, denoted by , represents the average number of customers that can be served per time period. The average service time per customer is 1/ time periods.

Service Rate

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Queuing Theory

1/2/3

1 The first characteristic identifies the nature of the arrival process using the following standard abbreviations:

M = Markovian interarrival times (following an exponential distribution)

D = Deterministic interarrival times (not random)

Kendall Notation

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Queuing Theory

2 The second characteristic identifies the nature of the service times using the following standard abbreviations:M = Markovian service timesG = General service times

(following a non-exponential distribution)

D = Deterministic service times (not random)

3 The third characteristic indicates the number of servers available.

Kendall Notation

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Queuing Theory

U - Utilization factor, or the percentage of time that all servers are busy.

P0 - Probability that there are no units in the system.

Lq - Average number of units in line waiting for service

L - Average number of units in the system (in line and being served)

Wq - Average time a unit spends in line waiting for service

T - Actual time a unit spends in the queueW - Average time a unit spends in the system

(in line and being served)Pw - Probability that an arriving unit has to

wait for servicePn - Probability of n units in the system

Operating Characteristics

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Queuing Theory

• There are s servers in the system, where s is a positive integer

• Arrivals follow a Poisson distribution and occur at an average rate of per time period

• Each server provides service at an average rate of per time period, and actual service times follow an exponential distribution

• Arrivals wait in a single FIFO queue and are serviced by the first available server

• < s

The M/M/s Model

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Queuing TheoryFormulas describing the M/M/s Model

Us

Pn s

ss

LP

s s

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WL

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s

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s

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1

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Queuing TheoryFormulas describing the M/M/s Model

=s

W W

Ps

ss

P

Pn

P for n s

Ps s

P for n s

p T ts

P e

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w

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Queuing TheoryQ.xls

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Queuing TheoryPrime Health is a large fictitious preferred provider organization (PPO) operating

in the Southwest. Like all PPOs, Prime Health realizes the importance of carefullymanaging the extent to which its primary care providers refer patients to specialists. PrimeHealth’s contract with their primary care physicians therefore requires care provider mustobtain authorization for the referral from the PPO.

The primary care physician typically obtains the required authorization by havingan office assistant telephone the PPO office, where a single authorization clerk is availableto handle the request. If the authorization clerk is busy with another call, then thesubsequent callers are placed on hold, and their calls are handled by the authorization clerkin the order in which they arrived. Because of the importance of receiving the requiredauthorization, the office assistant never hangs up before talking with the authorizationclerk. However, a survey conducted by the PPO among its provider network reveals thatextreme dissatisfaction results if the wait exceeds five minutes. Prime Health hasdetermined from a special study conducted during the past month that calls arrive at anaverage rate of 18 per hour. The same study reveals that the PPO operator requires anaverage of two minutes to process a given call. Thus the average service rate is 30 perhour.

Operation of such a system involves a number of costs: 1. The authorization clerk employed by the PPO earns a salary of $9.80 per hour

including benefits.2. The office assistant who places the telephone request for the authorization earns a

salary of $8.35 per hour including benefits.3. While the office assistant is on hold, he or she is unable to perform any additional

productive functions for the office.4. The cost associated with adding an additional telephone line to the PPO system is

$50.00 per month, including prorated installation costs.

For such a system, a number of questions could be raised. The physicians officesare quite interested in answering the following questions:

1. How often will the office assistant be placed on hold?2. How long will the office assistant need to wait until the call is handled?3. Will the office assistant ever receive a busy signal?

The PPO is also concerned with a number of issues. These might include:

1. The amount of time that the authorization clerk is idle;2. The improvement in service that results from adding a second clerk;3. The number of incoming lines needed to provide a particular level of service;4. The fraction of incoming calls when the time on hold exceeds the maximum tolerable

level of five minutes.

Case Problem (A) p. 140

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Queuing Theory

Case Problem (cont.)

M/M/s queuing computationsArrival rate 18 Assumes Poisson process for

Service rate 30 arrivals and se rvices.

Number of s ervers 1 (max of 40)

Utilization 60.00%P(0), probability tha t the system is empty 0.4000Lq, expected queue length 0.9000L, expected number in system 1.5000Wq, expected time in queue 0.0500W, expected total time in system 0.0833Probability that a customer waits 0.6000

00.20.4

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40NUMBER IN SYSTEM

PROBABILITY

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Queuing Theory


Probability that the Time in Queue is Greater than t forSelected Values of t (Basic Single-Server System)

t (minutes) p( T > t )

0 0.6 P 0 = 0.41 0.491238 s = 12 0.402192 = 303 0.329287 = 184 0.269597 = 0.65 0.2207286 0.1807177 0.1479588 0.1211389 0.099179

10 0.081201

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Queuing Theory


M/M/s queuing computationsArrival rate 18 Assumes Poisson process for

Service rate 30 arrivals and se rvice s.

Number of s ervers 2 (max of 40)

Utiliza tion 30.00%P(0), probability that the system is empty 0.5385Lq, expected queue length 0.0593L, expected number in system 0.6593Wq, expected time in queue 0.0033W, expected tota l time in system 0.0366Probability that a customer waits 0.1385

00.20.40.6

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40NUMBER IN SYSTEM

PROBABILITY

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Queuing Theory

Probability that the Time in Queue is Greater than t forSelected Values of t (Basic Two-Server System)

t (minutes) p( T > t )

0 0.138471 P 0 = 0.53851 0.068763 s = 22 0.034147 = 303 0.016957 = 184 0.00842 = 0.35 0.0041816 0.0020767 0.0010318 0.0005129 0.000254

10 0.000126


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Queuing TheoryFinite Queue Model

M/M/s with Finite Queue Arrival rate 18 Service rate 30 Number of s ervers 1 (max of 40)

Maximum queue length 5 (max of 40 c ombine d)

Utiliza tion 58.85%P(0), probability that the system is empty 0.4115Lq, expected queue length 0.7099L, expected number in system 1.2984Wq, expected time in queue 0.0394W, expected tota l time in system 0.0728Probability that a customer waits 0.5885Probability that a customer balks 0.0192

00.20.40.6

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40NUMBER IN SYSTEM

PROBABILITY


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Queuing TheoryFinite Queue Model

M/M/s with Finite Queue Arrival rate 18 Service rate 30 Number of s ervers 2 (max of 40)

Maximum queue length 5 (max of 40 c ombine d)

Utilization 29.99%P(0), probability that the system is empty 0.5385Lq, expected queue length 0.0587L, expected number in sys tem 0.6586Wq, expected time in queue 0.0033W, expected total time in system 0.0366Probability that a customer waits 0.1384Probability that a customer balks 0.0002

00.20.40.6

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40NUMBER IN SYSTEM

PROBABILITY


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Queuing Theory

No. of Clerks

No. of Hold Positions p(Busy)

Exp. Time in Queue

(minutes)Utilization

Factor

1 0 0.375 0 0.3751 1 0.184 0.75 0.4901 2 0.099 1.35 0.5401 3 0.056 1.81 0.5661 4 0.033 2.16 0.5801 5 0.019 2.41 0.5881 6 0.011 2.6 0.5931 7 0.007 2.73 0.5961 8 0.004 2.82 0.5981 9 0.002 2.88 0.5991 infinite 0 3 0.6002 0 0.1011 0 0.2702 1 0.0294 0.101 0.2912 2 0.0088 0.157 0.2972 3 0.0026 0.182 0.2992 4 0.0008 0.192 0.3002 5 0.0002 0.196 0.3002 6 0.0001 0.197 0.3002 infinite 0 0.198 0.3003 0 0.0198 0 0.1963 1 0.0039 0.013 0.1993 2 0.0008 0.018 0.2003 3 0.0002 0.02 0.2003 infinite 0 0.021 0.200


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Queuing Theory

Configuration

Monthly Cost of Lines

Monthly Cost of Clerks

Exp. No. in System

Monthly Cost of Waiting

Total Monthly

Cost

1 Clerk, 5 hold positions $300 $1,699 1.2984 $1,879 $3,8782 Clerk, 2 hold positions 200 3,397 0.6414 928 4,5253 Clerk, 0 hold positions 150 5,096 0.5881 851 6,097

Authorization clerk's monthly salary = $9.80/h x 40 h/week x 4.333 weeks/month = $1,698.66Office assistant's monthly salary = $8.35/h x 40 h/week x 4.333 weeks/month = $1,447.33


Queuing Theory

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