QUEUING MODELS

QUEUING MODELS

Based on slides for Hilier, Hiller, and Lieberman, Introduction to Management Science, Irwin McGraw-Hill

Queuing Theory

Waiting occurs in

Service facility Fast-food restaurants post office grocery store bank

Manufacturing

Equipment awaiting repair

Phone or computer network

Product orders

Why is there waiting?

System Characteristics

Number of servers

Arrival and service pattern

Queue discipline

Measures of System Performance

Average number of customers waiting

Average time customers wait

System utilization

Number of ServersSingle Server . . .

Customers ServiceCenter

Multiple Servers

. . .

Customers

ServiceCenters

Multiple Single Servers

. . .

. . .

. . .

Customers ServiceCenters

Some Assumptions

Arrival Pattern: Poisson

Service pattern: exponential

Queue Discipline: FIFO

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Cu stom ers p er time u nit

.02

.04

.06

.08

.10

.12

.14

.16

.18

Relative Frequency

Service Time

Relative Frequency (%)

. . .

CustomersServiceCenter

Some Models

1. Single server, exponential service time (M/M/1)

2. Multiple servers, exponential service time (M/M/s)

A TaxonomyA / B / s

Arrival Service Number ofDistribution Distribution Servers

whereM = exponential distribution (“Markovian”)D = deterministic (constant)G = general distribution

Given

= customer arrival rate = service rate (1/m = average service time)s = number of servers

Calculate

Lq = average number of customers in the queue

L = average number of customers in the system

Wq = average waiting time in the queue

W = average waiting time (including service)

Pn = probability of having n customers in the system

= system utilization

Model 1: M/M/1 Example

The reference desk at a library receives request for assistance at an average rate of 10 per hour (Poisson distribution). There is only one librarian at the reference desk, and he can serve customers in an average of 5 minutes (exponential distribution). What are the measures of performance for this system?

M/M/s Queueing Model Template

Data 10 (mean arrival rate) 12 (mean service rate)s = 1 (# servers)

Prob(W > t) = 0.135335when t = 1

Prob(Wq > t) = 0.112779

0 when t = 1

ResultsL = 5 Number of customers in the system

Lq = 4.166666667 Number of customers in the queue

W = 0.5 Waiting time in the systemWq = 0.416666667 Waiting time in the queue

0.833333333 Utilization

P0 = 0.166666667 Prob zero customers in the system

Model 2: M/M/s Example

The Federal Bank of Washington has three tellers at their Seattle branch. Customers arrive randomly at the branch at an average rate of 1 per minute. The service time averages 2 minutes, and follows the exponential distribution. What are the measures of performance?



Prob(W > t) = 1.34E-12when t = 1

Prob(Wq > t) = 8.32E-14

0 when t = 1

ResultsL = 2.888888889 Number of customers in the system





Application of Queuing Theory

We can use the results from queuing theory to make the following types of decisions:

How many servers to employ

Whether to use one fast server or a number of slower servers

Whether to have general purpose or faster specific serversGoal: Minimize total cost = cost of servers + cost of waiting

Cost ofService Capacity

Cost of customerswaiting

Total Cost

OptimumService Capacity

Cost

Example #1: How Many Servers?

In the service department of an auto repair shop, mechanics requiring parts for auto repair present their request forms at the parts department counter. A parts clerk fills a request while the mechanics wait. Mechanics arrive at an average rate of 40 per hour (Poisson). A clerk can fill requests in 3 minutes (exponential). If the parts clerks are paid $6 per hour and the mechanics are paid $18 per hour, what is the optimal number of clerks to staff the counter.

Economic Analysis of M/M/s Queueing Model

Data 40 (mean arrival rate) Results 20 (mean service rate) L = 2.88888889 Number of customers in the systems = 3 (# servers) Lq = 0.88888889 Number of customers in the queue

Pr(w>t) = 0.001273 W = 0.07222222 Waiting time in the systemwhen t = 1 Wq = 0.02222222 Waiting time in the queue

Prob(wq>t) = 0.000848 0.66666667 Utilization

when t = 1P0 = 0.11111111 Prob zero customers in the system

Cs = 6.00$ (cost/server/unit time)

Cw = 18.00$ (waiting cost/unit time)

Cost of Service = 18.00$ Cost of Waiting = 52.00$

Total Cost = 70.00$



Pr(w>t) = 4.54E-05 W = 0.05434783 Waiting time in the systemwhen t = 1 Wq = 0.00434783 Waiting time in the queue

Prob(wq>t) = 2.27E-05 0.5 Utilization





Total Cost = 63.13$

So s = 4 has the smallest total cost.



Pr(w>t) = 6.14E-06 W = 0.05099502 Waiting time in the systemwhen t = 1 Wq = 0.00099502 Waiting time in the queue

Prob(wq>t) = 2.46E-06 0.4 Utilization





Total Cost = 66.72$

Example #2: How Many Servers?

Beefy Burgers is trying to decide how many

registers to have open during their busiest time,

the lunch hour. Customers arrive during the lunch

hour at a rate of 98 customers per hour (Poisson

distribution). Each service takes an average of 3

minutes (exponential distribution). Management

would not like the average customer to wait longer

than five minutes in the system

. How many registers should they open?



Prob(W > t) = 0.142902when t = 1

Prob(Wq > t) = 0.135226

0 when t = 1




0.98 Utilization


For five servers

Choose s = 6 since W = 0.0751 hour is less than 5 minutes.

For six servers



Prob(W > t) = 1.19E-08when t = 1

Prob(Wq > t) = 2.77E-10

0 when t = 1






Example #3: One Fast Server or Many Slow Servers?

Beefy Burgers is considering changing the way that they serve

customers. For most of the day (all but their lunch hour), they

have three registers open. Customers arrive at an average

rate of 50 per hour. Each cashier takes the customer’s order,

collects the money, and then gets the burgers and pours the

drinks. This takes an average of 3 minutes per customer

(exponential distribution). They are considering having just

one cash register. While one person takes the order and

collects the money, another will pour the drinks and another

will get the burgers. The three together think they can serve a

customer in an average of 1 minute. Should they switch to one

register?

3 Slow Servers

1 Fast Server

W is less for one fast server, so choose this option.

Data 50 (mean arrival rate) 20 (mean service rate) Resultss = 3 (# servers) L = 6.011235955 Number of customers in the system


Prob(W > t) = 6.38E-05when t = 1 W = 0.120224719 Waiting time in the system

Wq = 0.070224719 Waiting time in the queueProb(Wq > t) = 4.34E-05

0 when t = 1 0.833333333 Utilization


Data 50 (mean arrival rate) 60 (mean service rate) Resultss = 1 (# servers) L = 5 Number of customers in the system


Prob(W > t) = 4.54E-05when t = 1 W = 0.1 Waiting time in the system

Wq = 0.083333333 Waiting time in the queueProb(Wq > t) = 3.78E-05

0 when t = 1 0.833333333 Utilization


Example 4: Southern RailroadThe Southern Railroad Company has been subcontracting for painting

of its railroad cars as needed. Management has decided the company might save money by doing the work itself. They are considering two alternatives. Alternative 1 is to provide two paint shops, where painting is to be done by hand (one car at a time in each shop) for a total hourly cost of $70. The painting time for a car would be 6 hours on average (assume an exponential painting distribution) to paint one car. Alternative 2 is to provide one spray shop at a cost of $175 per hour. Cars would be painted one at a time and it would take three hours on average (assume an exponential painting distribution) to paint one car. For each alternative, cars arrive randomly at a rate of one every 5 hours. The cost of idle time per car is $150 per hour.

Estimate the average waiting time in the system saved by alternative 2.

What is the expected total cost per hour for each alternative? Which is the least expensive?Answer: Alt 2 saves 1.87 hours. Cost of Alt 1 is: $421.13 / hour and cost of Alt 2 is $400.06 /hour.

Answer: Alt 2 saves 1.87 hours. Cost of Alt 1 is: $421.13 / hour and cost of Alt 2 is $400.06 /hour.

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QUEUING MODELS