Waiting Line Models

(Queuing Theory)

Lines in Operations Management

Assembly lines

Production lines
Trucks waiting to unload or load Workers waiting for parts Customers waiting for products Broken equipment waiting to be fixed Customers waiting for service

2012 Lew Hofmann

Costs
The cost of waiting

Losing customers because of long lines

Reneging: Customers get tired of waiting and leave Balking: Customers see a long line and dont get in line.

Paying employees to wait for something they need.

(waiting for parts, supplies, deliveries, etc.)

Unusable (idle) equipment awaiting repairs

EG: Broken assembly line machinery.

The cost of service

Paying people to provide service to customers

Customers can be people, machines, or other objects needing service.

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Cost of Providing Service

Paying repairmen to fix broken machines Paying dock workers to load and unload trucks Paying customer-service people Using more production people to speed up the line Leasing of service equipment and facilities Paying checkout cashiers
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Queuing System Costs

Costs

Costs of Waiting
Fewer servers often means longer waiting for customers.

goes down as service improves.

Optimal # of servers

Number of Servers

Many servers means little or no waiting, but higher service costs.

Note that the lowest cost system requires some customer waiting.
2012 Lew Hofmann

 Average time in line for a customer. Average number of customers in line. Average time in the system for a customer. Average number of customers in the system

What Queuing Models Tell Us.

Probability of n number of customers in the

system at any given time.

at any time.

NOTE: In The System includes customers who

are in line plus the customers being served.
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The Waiting Line System

ARRIVAL SYSTEM
(How customers arrive)

QUEUE
(The nature of the waiting line or lines of customers)

SERVICE FACILITY
(How customers progress through the service facility)
2012 Lew Hofmann

Waiting Line Models

Customer population

Service System
Arrival System

Waiting line (Queue)

Priority rule

Service facilities

Served customers

The sequence in which customers are admitted into the service facility.

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Arrival System

Arrival Populations are either

Arrival Patterns are either Behavior of the Arrivals

Limited (EG: Only people age 21 or over.) Unlimited (EG: cars arriving at a toll booth) Random (Each arrival is independent) Scheduled (EG: Doctors office visits) Balking (Seeing a long line and avoiding it.) Reneging (Get tired of waiting and leave the line) Jockeying (Switching lines)

2012 Lew Hofmann

The Queue

Queue Length is either..

Queue Discipline (order of service)

FIFO (First-In, First-Out) LIFO (Last-In, First-Out) SIRO (Service In Random Order) Priority

Unlimited (EG: cars in line at a toll booth)

Limited (Finite) EG: # of e-mail messages allowed.

2012 Lew Hofmann

The Service Facility

How many paths (ways to get through the

system) are there after getting in line? EG: McDonalds drive-thru is one channel.

Channels

How many stops must a customer make,

after getting in line?
(Single-phase means only one stop for service.) McDonalds drive-thru is a three-phase system: 1. Order 2. Pay 3. Pick-up

Phases

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Single-channel, Single-phase
One way through the system and one stop for service
Service Facility

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Multi-channel, Single-phase
Once in line, you have at least two choices of how to get through the system, but only one stop. Service Facility Service Facility

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Multi-channel, Multi-phase
Once in line, you have at least two choices (channels) of how to get through the system and at least two stops (phases).

Service Facility Service Facility

Service Facility Service Facility

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Four Single-channel, Single-phase Systems

(Once in line, you only have one channel and one stop.) Service Facility Service Facility Service Facility Service Facility
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One, Multi-channel, Single-Phase System

(Once in line you have four possible paths through the system, but only one stop.)

Service Facility
Service Facility Service Facility Service Facility
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Assumptions of our Models

The Rate of Service must be faster than the Rate of

FIFO (First In, First Out) (Customers are served in the order they arrive.) Arrivals are unlimited (infinite) Arrivals are random rather than scheduled. Customers arrive independently of each other. Service times can vary from one customer to
another, and are independent of each other.
(Customers may have different service needs and times.)

Arrivals. (It is unsolvable if customers arrive faster than they can be served.)

2012 Lew Hofmann

Queuing Problem
At a large Naval Ship Repair Facility mechanics have to make frequent trips to the tool crib for parts and specialized equipment. (Arrivals are infinite since mechanics
can come as often as need, even though the population of customers is finite.)

Records indicate that the tool crib serves an average of 18 mechanics each hour, but is capable of serving 20 per hour. If mechanics are paid $30 per hour and the tool crib attendants make $9 per hour, would it be more cost effective to have one or two attendants in the tool crib?
The service rate is always the average time for one server, regardless of how many servers there are in the system. Here it is 20, which is higher than the arrival rate of 18. If the service rate had been lower than the arrival rate, the problem would not be solvable, because customers would arrive faster than they could be served.
2012 Lew Hofmann

Which system is less expensive?

Single Attendant

1st Attendant
2nd Attendant
(It depends on the relative costs of service versus waiting.)

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Data Entry Information for POM/QM

One attendant can serve 20 mechanics/customers per hr. and is paid $9 per hour. Single Attendant Mechanics (customers) arrive at an average of 18 per hour and are paid $30 per hour. 1st Attendant

2nd Attendant
I ran the POM-QM model using two servers, but I could have run it with any number of servers since you always enter the service rate for one server. The POM-QM model will do the computations for more than one server.
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Service rate for 1 server!

Input data

Lowest Cost!

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Probability that the system is idle (no customers) is 37.93%

The probability that 4 or fewer mechanics are in the system is 97.45%

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What the Model Tells Us

Note: You need to run the model with 1 server if you want the info for 1 server, and run it using 2 servers to get the info for 2 servers.

# of Servers 1 2

Average # customers in the system 9.0 Average time in the system Average # customers in line Average time in line

1.128

0.5
8.1

0.063
0.228

0.45 0.013
0.1 0.38

Probability that the system is idle

Once you know the optimal # of servers, make sure you run it again for that many servers in order to get the right data. But always enter the service rate for one server, regardless of how many servers.
2012 Lew Hofmann

Make sure you use the M/M model.

Results for 2 servers

Average # customers in the system Average time spent in the system Average # of customers in line Average time in line
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POM/QM or Excel Solver?

POM/QM will do the cost analysis for you, so it is easier. Select the M/M model. You can use the Excel Solver but it wont do the cost analysis. You also need to run the Excel solver once each server number you use, and then do a manual cost analysis for each in order to see which number of servers has the lowest cost. In multiple server problems, you might have to run it for a half-dozen or more scenarios.
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Excel Solver: 1 Server

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 Attendants get paid $9 per hour

Manual cost computations for one server (9 customers in the system)

Mechanics get paid $30 per hour

(Cost of service is thus 1 x $9 = $9 per hour)

Total cost of one server is: $270 + $9 = $279

POM-QM model

Cost of waiting is: $30 x 9 customers in system = $270

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Excel Solver: Two Servers

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Manual cost computations for Two servers (9 customers in the system)

Attendants get paid $9 per hour
Cost of service is: $9 x 2 servers = $18 per hour

Mechanics get paid $30 per hour

Cost of waiting is: $30 x 1.28 customers in system = $33.84

Total cost of two servers is: $18 + $33.84 = $51.84

POM-QM model

2012 Lew Hofmann

Homework Assignment
Due next Tuesday Problem 1: Car Repair Problem 2: The Quarry
Use POM/OM or Excel Solver software and submit printouts to support your decisions

2012 Lew Hofmann

Car Repair
In the service department of a car repair shop, mechanics requiring parts for auto repair or service present their request forms at the parts department counter. The parts clerk fills a request while the mechanic waits. Mechanics arrive in a random fashion at the rate of 40 per hour, and a clerk can fill requests at the rate of 20 per hour. If the cost for a parts clerk is $6 per hour and the cost for a mechanic is $12 per hour, determine the optimum number of clerks to staff the counter. (Because of the high arrival rate, an infinite source may be assumed.)
Always enter the service rate for one server. The program will do the math once you enter the number of servers. If you enter fewer servers than can handle the arrival needs, the program will give you an error message because the computed service rate must be higher than the arrival rate.

2012 Lew Hofmann

The Quarry (no cost analysis in model)

You are in charge of a quarry that supplies sand and stone aggregates to construction sites. Empty trucks arrive at the quarry and wait in line for loading either sand or aggregate. At the loading station they are filled with material, weighted, checked out, and then proceed to a construction site.

Currently 9 empty trucks arrive each hour (on average). In addition to waiting in line, it takes 6 minutes for a truck to be filled, weighed and checked out.
Concerned that trucks are spending too much time waiting and being filled, you evaluate the current situation and compare it to the 2 alternatives below.

Alternative 1: Speed up the loading process and add side boards to the trucks so that more material can be loaded faster. This will improve the speed of loading, but cost $50,000. Since the trucks hold more, their arrival rate would be reduced to 6 per hour and the loading time would be reduced to 4 minutes each.
Alternative 2: Add a second loading station at a cost of $80,000. The trucks would arrive at the current rate of 9 per hour. They would then wait in a common line and the truck at the front of the line would move to the next available loading station. Loading time at each of the stations is 6 minutes. Which alternative do you recommend? (Select No Cost in the POM/QM waiting line model. You must decide which of the three situations would the most cost effective based on time in the system and upgrade costs.
2012 Lew Hofmann

Facts About Queuing

QUEUE originally (15th century) referred to the tail of the beast or a tail piece. (Not to be confused with a piece of tail.) In the 17th century a queue became a braid of hair. Later it was used to refer to a pigtail.

In the 18th century a billiard stick became a queue, later changed to que and then to cue.
In the early 19th century England, to Queue was to line up. And that is how it is used today in England.

2012 Lew Hofmann