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Queuing Problems

1. Sharp Discounts Wholesale Club has two service desks, one at each
entrance of the store. Customers arrive at each service desk at an
average of one every six minutes. The service time at each service desk is
four minutes per customer.

a) How often (what percentage of time) is each service desk idle?
b) What is the probability that both service clerks are busy?
c) What is the probability that both service clerks are idle?
d) How many customers, on average, are waiting in line in front of each service
desk?
e) How much time does a customer spend at the service desk (waiting plus
service time)?

2. Burrito King (a new fast food franchise opening up nationwide) has
successfully automated burrito production for its drive-up fast food
establishments. The Burro-Master 9000 requires an average service time of
45 seconds, exponentially distributed, to produce a batch of burritos. It has
been estimated that customers will arrive at the drive-up window according to a
Poisson distribution at an average of one every 50 seconds. To help determine
the amount of space needed for the line at the drive-up window, Burrito
King would like to know the expected average time in the system, the
average line length (in cars), and the average number of cars in the system
(both in line and at the window).

3. The Bijou Theater in Hermosa Beach, California, shows vintage movies.
Customers arrive at the theater line at the rate of 100 per hour. The ticket
seller averages 30 seconds per customer, which includes placing validation
stamps on customers’ parking lot receipts and punching their frequent
watcher cards. (Because of these added services, many customers don’t
get in until after the feature has started.)

a) What is the average customer time in the system?

b) What would be the effect on customer time in the system of having a second
ticket taker doing nothing but validations and card punching, thereby cutting the
average service time to 20 seconds?
c) Would system waiting time be less than you found in b) if a second window
was opened
with each server doing all three tasks?

Where Does Variability Exist In Operations?

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Where Does Variability Exist In Operations?

Demand

Operation Completion

Transport

Labor

Machine Downtime

Raw Material Quality

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How Does Variability Affect Operations?

Capacity?

Utilization?

Efficiency?

Throughput Time?

Cost?

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Tools For Capacity Planning With Variability

Without Variability – Long Division

With Variability:

Linear Programming – to allocate capacity over multiple facilities or multiple time periods

Simulation – model any complex system using random numbers

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New Tool:
Queueing Models

Mathematical analysis of waiting line systems of all types

Fast

Little data needed

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Why Do Queues Exist?

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Why Do Queues Exist?

They are due to short run variations in demand and capacity

To understand how much additional capacity we need, we need to understand queues

But are all queues the same?

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Elements of A Queueing System

Customers (Orders, Raw or WIP Materials, etc.) arrive according to a known or predictable random process – Arrival (or birth) Process

Arrivals wait (literally or figuratively) in line to be serviced – Queue

Arrivals are selected out of the line according to a rule such as first come first serve – Queue Discipline

Customers enter service from queue then exit system when service is complete – Service (or Death) Process

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Elements of A Queueing System

Queue Structure

Single Line

Single Queue/Single Server or

Single Queue/Multiple Server

Parallel Lines – Multiple Single Queue/Single Server Lines

Stages Service – A Sequence of operations each with a queue

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Elements of A Queueing System

Service Order

First Come First Served

Last Come First Served

Random Order

Shortest Processing Time First

Priorities

Appointments/Due Dates

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Data/Facts You Need For A Queueing Model

Customer Arrival Rate – Average number of customers arriving per time period

Service Time – Average time a server takes to service a customer

Service Rate – Average number of customers a server can serve per unit of time

Number of Servers

Queue Structure

Queue Discipline

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Random Arrival Process

Poisson Process

Arrivals are independent of one another

Average arrival rate is constant over period of analysis

Arrivals occur one at a time

Number of arrivals in an interval is described by a Poisson distribution

The time between arrivals is described by an Exponential Distribution

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Definitions

λ – Average arrival rate: The number of customers arriving per unit time

μ – Average service rate: The number of customers a server can serve per unit time

– Average service time: Average time to serve one customer by one server

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Average Time Spent In The Queue

Average Time In System (Time in queue plus time in service)

Average Length Of Queue

Average Number Of Customers In The System

Probability That A Customer Waits Before Service Begins (Probability of a Delay)

Server Utilization

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Some Performance Measures

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Simple Formulae For Simple Case

Single Queue/ Single Server/FCFS

Utilization () Equals average arrival rate over average service rate

Average Time In The System

Average Number In System

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Formula… cont’d.

Average Time In The Queue

Average time in system minus average time in service

Average Number In The Queue

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Multiple Servers

Formulae Are More Complicated

Utilization () Equals average arrival rate over number of servers (S) times average service rate

All performance measures can be obtained by using MMs Excel spreadsheet on Courseworks

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Multiple Servers/Single Queue Formulae

Find Lq from Table

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Example

The manager of a store has two clerks in different parts of store. λ of 15 Overall

Same as two single server queues each with λ of 7.5 customers per hour

μ = 15 per Hour

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Measures For This Alternative:
2 Single Queue Single Server

For Each

Number in Queue:

Number in System: customer at each place

Time in System: hour or 8 minutes

Time in Queue: hour or 4 minutes

For Entire System

Time in System = 8 minutes

Number in System = 2 customers

Time in Queue = 4 minutes

Number in Queue = 1 customer

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Multiple Server / Single Queue

Put the two clerks together

Have customers form one line that feeds the two servers

Calculate measures of performance

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Multiple Servers Measures:
Single Queue Two Servers

λ = 15 per Hr

μ = 15 per Hr

Number of Servers = 2

Utilization = 15/2*15 = 50% – Same

Number In Queue: Lq = From Table for 15/15 and 2 Servers = 0.333 cust.- Better

Time In Queue = Wq = 0.333/15 Hrs = 1.33 min – Better

Number In System = Ls = 0.333+(15/15) = 1.333 – Better

Time In System = Ws = 1.333/15 Hrs or 5.33 min. – Better

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Another Example

The Manager of the Store Has Another Choice. He Can Have:

One Fast Check Out Clerk Who Can Process One Customer Every Two Minutes or

Two Moderately Fast Clerks Who Can Each Service A Customer In Four Minutes.

Which Alternative Is Better? (Assume Customers Arrive Randomly at the Rate of 15 per Hour in Each Case.)

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Single Server Measures:
Single Queue Single Server

λ = 15 per Hour

μ = 30 per Hour (One Every Two Min.)

Utilization = 15/30 = 50%

Time In System = 4 Minutes

Average Number in System = 1 Customer

Average Wait in Queue = 2 Minutes

Average Number in Queue = 0.5 Customers

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Multiple Servers Measures:
Single Queue Two Servers

λ = 15 per Hr

μ = 15 per Hr

Number of Servers = 2

Utilization = 15/2*15 = 50% – Same

Time In System = 5.33 min. – Worse

Number In System = 1.33 cust. – Worse

Time In Queue = 1.33 min. – Better

Number In Queue = 0.33 cust. – Better

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Mathematical Lessons About Queueing Theory

λ must be less than μ per server times the number of Servers ()

If λ is well below , waiting times are low – Customer is happy, but utilization is low – boss sees waste

If λ is close to , waiting times are high – customer is unhappy, but utilization is high – workers are busy

For multiple servers, better to have single queue

If not FIFO, order by expected time of completion – good for most but bad for the longest

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More Lessons

For series or networks, look for weakest link, balance network

In low utilization systems (Emergency Response) find something else for them to do

Special queues for special needs

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mms

mms.xls M/M/s Queueing Formula Spreadsheet
Inputs: Definitions of terms:
lambda 0.1 lambda = arrival rate
mu 0.3 mu = service rate
s = number of servers
Lq = average number in the queue
Ls = average number in the system
Wq = average wait in the queue
Ws = average wait in the system lambda/mu
P(0) = probability of zero customers in the system 0.33333333333333337
P(delay) = probability that an arriving customer has to wait
Outputs: Intermediate Calculations:
s Lq Ls Wq Ws P(0) P(delay) Utilization (l/u)^s/s! sum (l/u)^s/s!
0.0 1.0 1.0
1.0 0.1667 0.5000 1.6667 5.0000 0.6667 0.3333 0.3333 0.33333333333333337 1.3333333333333335
2.0 0.0095 0.3429 0.0952 3.4286 0.7143 0.0476 0.1667 0.055555555555555566 1.388888888888889
3.0 0.0006 0.3340 0.0062 3.3396 0.7164 0.0050 0.1111 0.006172839506172842 1.395061728395062
4.0 0.0000 0.3334 0.0004 3.3337 0.7165 0.0004 0.0833 5.144032921810702E-4 1.395576131687243
5.0 0.0000 0.3333 0.0000 3.3334 0.7165 0.0000 0.0667 3.429355281207135E-5 1.3956104252400552
6.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0556 1.9051973784484087E-6 1.3956123304374337
7.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0476 9.072368468801947E-8 1.3956124211611185
8.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0417 3.7801535286674784E-9 1.395612424941272
9.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0370 1.4000568624694365E-10 1.3956124250812778
10.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0333 4.666856208231456E-12 1.3956124250859447
11.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0303 1.4141988509792291E-13 1.3956124250860862
12.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0278 3.928330141608971E-15 1.3956124250860902
13.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0256 1.0072641388740952E-16 1.3956124250860902
14.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0238 2.398247949700227E-18 1.3956124250860902
15.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0222 5.329439888222727E-20 1.3956124250860902
16.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0208 1.1102999767130683E-21 1.3956124250860902
17.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0196 2.1770587778687616E-23 1.3956124250860902
18.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0185 4.031590329386596E-25 1.3956124250860902
19.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0175 7.072965490151924E-27 1.3956124250860902
20.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0167 1.1788275816919872E-28 1.3956124250860902
21.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0159 1.871154891574583E-30 1.3956124250860902
22.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0152 2.8350831690523986E-32 1.3956124250860902
23.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0145 4.108816187032462E-34 1.3956124250860902
24.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0139 5.706689148656198E-36 1.3956124250860902
25.0 0.0000 0.3333 0.0000 3.3333 0.7165 0.0000 0.0133 7.608918864874932E-38 1.3956124250860902

Queuing Models ( A = arrival rate; p = service rate)

Mfl/1 Queue

++++.) –
Anivals

Expected number in system: A L =-
P – A

Expected number in queue: L, =
A

P(P – A)
1 Expected waiting time (includes service time): W = –

C L – A

Expected time in queue:

Probability that the system is empty:

M/G/l Queue

+ + + + + –
Arrivals

Expected number in system:

Expected number in queue: L, =
AZcr2 + (Alp)’

2(1 – Alp)
I

Expected waiting time (includes service time): W = Wq + –
CL

L
Expected time in queue: Wq = A

Probability that the system is empty:

M ! / s Queue

Arrivals

Expected number in system:

Expected number in queue:

Expected waiting time
(includes service time):

Expected time in queue:

probability that the system is empty:

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