Department of Systems Engineering and Engineering ManagementCharles V. Schaefer Jr. School of EngineeringStevens Institute of TechnologyHoboken, New Jersey 07030

Queuing

Example of Queuing Systems

SYSTEM CUSTOMERS SERVER(S)

Reception Desk People ReceptionistRepair Facility Machines RepairpersonGarage Trucks MechanicTool Crib Mechanics Tool Crib ClerkHospital Patients NursesWarehouse Pallets CraneAirport Airplanes RunwayProduction Line Cases Case PackerRoad Network Cars Traffic LightMarket Shoppers Checkout CounterLaundry Dirty Linen Washing Machines/DryersJob Shop Jobs Machines/WorkersLumberyard Trucks Overhead CraneSaw Mill Logs SawsComputer Jobs CPU/Disk/TapesTelephone Calls ExchangeTicket Office Football Fans ClerkMass Transit Riders Buses/Trains

Definitions

Queue waiting line

Queuing Systems system involving a queue

Queuing Theory techniques to describe the

behavior of a queuing systems

Calling Units members of the the queue

Service Facility the source of the goods or

services for which the calling units wait

Single Channel only one server

Multi Channel more than one server

performing the same service

CALLINGPOPULATION QUEUE

SERVICEFACILITY

SERVEDCALLING

UNITS

-SIZE -LENGTH -STRUCTURE-ARRIVAL PATTERN -DISTRIBUTION-ATTITUDE -DISCIPLINE

Queuing System

Queuing Terms and Statistics

Lq = mean number of calling units in the queue

L or Ls = mean number of calling units in the system

Wq = mean time spent in the queue

W or Ws = mean time spent in the system

The subscript q means

queue whereas the s means

the system or the queue plus

the server

Queuing Terms and Statistics

= Mean arrival rate (#units/unit time)

1/ = Interarrival time (time between units)

= Mean service rate (#units served/unit time)

1/ = Mean service time (time/unit)

Mu and Lambda are always

rates the inverse of a rate

is a time for example

customers/min is the rate

and the time would be

minutes/cust

Queuing Terms and Statistics

S = Number of equivalent servers in the

service facility

U = Server utilization (% of time server

expected to be busy)

P(n) = Probability of having n units in the system

M = Maximum number of calling units in

the system (finite or limited source

problems)

Simple Stream

Stationary Mean is the same for any interval of measurement

No Aftereffects Number of occurrences in an interval is independent of number of occurrences in any other interval

(Markovian Property)

Orderliness Probability of more than 1 event approaches zero a interval approaches zero (i.e. for very small h)

Theorem:

A Simple Stream (Stream with above properties) will result in an arrival pattern that follows the Poisson Distribution)

Poisson Arrival Times Distribution

Poisson Arrival (probability) distribution:

Let x be a random variable of the number of arrivals

Let p(x) = probability of x arrivals in time t and let denote the

average arrival rate in time t

If the arrival distribution is Poisson then:

( ) ( ) / !n t

tP x n t e n

0,1, 2, ....n

Where = # of arrivals in interval of time t = Average arrival rate in interval t( )

tP x n is probability of exactly n arrivals over interval t

0( 0) ( ) / 0 !

t t

tP x t e e

( 0) 1

t

tP x e

n

Queuing Formulas

Basic Single ServerM/M/1 : Poisson Arrivals, Exponential service times, 1 Server

0

2

1 /

( ) (0)( / )

/ ( )

/ ( )

/ ( )

1 / ( )

/

n

q

P

P n P

Lq

Ls

W

Ws

Arrival Distribution

Arrivals Rate - Poisson

0

1000

2000

3000

4000

5000

6000

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Arrivals Per Day

Fre

qu

en

cy

Series1

= 4 Arrivals/day

Interarrival Time Distribution

Interarrival Distribution IA

0

5000

10000

15000

20000

25000

30000

0 2 4 6 8 10 12 14 16 18 20

Interarrival Time

Fre

qu

en

cy

Series1

Minutes x 100

Avg. IAT = = .25 days = 360 Minutes1 /

Basic Single Server

Poisson Arrival Process

Exponential Service Times

Single Server

FCFS Server Discipline

Infinite Source

Infinite Queue

Basic Multiserver

Same as Basic Single Server Except

Multiple Servers

Types of Models and Assumptions

Our main focus in this

class will be the basic

single server

Types of Models and Assumptions

Single Server with Arbitrary Service TimesPoisson Arrival ProcessService Time Distribution is Unknown(has a mean and variance)Single ServerFCFS Server DisciplineInfinite Source, Infinite Queue

Single Server Model with Arbitrary Service Times and aPriority Queue DisciplinePoisson Arrival ProcessService Time for Each Priority Class is UnknownSingle ServerQueue Discipline Divided Into Classes; Service is FCFS Within Each Priority Class Infinite Source, Infinite Queue

Server Utilization

s

non

linear

Single Server

Multi Servers

Server utilization is important when looking at the

efficiency of a system

Queuing Results

All Queuing Statistics in our closed form analytical models

are STEADY STATE statistics

Steady State: Steady State conditions exist when a

systems behavior is not a function of time.Transient: Startup Phase

Time

f(x)

Steady State

Startup

A Queuing Problem Solving

Methodology

Read the Problem!!! Picture/Sketch the system components Identify the underlying assumptions for each component Identify the model based on the assumptions Identify , , s (ensure its a RATE, time units the same) Determine WHAT statistic(s) you have to solve for Crunch

Class Problem 2-1

New Jersey Transit (NJT) operates a single train wash as

part of its Hoboken facility. The service time of this

machine is 30 minutes per train. The arrival time of trains

at the cleaning facility can be approximated as a Poisson

process with an arrival rate of 1.4 trains per hour. The

washing center operates 24 hours a day.

a. State the queuing model most appropriate for this

situation (assume that NJT owns and operates a lot of

trains).

b. How many trains on average are waiting to use the

facility?

c. How much time will each train spend at the facility (in

line and being washed)?

d. What is the utilization rate of the washing facility?

e. Do we need to investigate in building another washing

facility or purchasing more efficient equipment?

Quantitatively justify your answer!

Problem 2.1

A. Basic Single Server (trains always running i.e assume infinite pop)

B. How many trains on average are waiting to use the facility?

C. How much time will each train spend at the facility (in line and washed)?

D. What is the utilization rate of the washing facility?

E. Do we need to investigate another washing facility or purchase more efficient equipment?

Utilization rate would drop to 35% (not efficient). More efficient equipment is probably the way to go.

1.4 trains per hour

2 trains per hour

1 /( ) 1 / .6 1.67Ws Hours

/ 1.4 / 2 .7 70%

2/ ( ) 1.96 / 2(.6) 1.63

qL trains

Class Problem 2-3

As a newly assigned manger of Home Depot, you have received complaints about

the checkout time at the lawn and garden center. You are currently operating one

cash register. You decide to do a queuing analysis on the center and assume the

following:

- The calling population is infinite

- The pattern of arrivals is Poisson distributed

- The customers are patient and will not balk or renege

- The queue length is unlimited (customers will join the system no

matter how long the line is or how crowded the store is).

- The service times by the cashier are exponentially distributed

- The service discipline is first come, first served (no cutting in line by

senior ranks)

Your cash register information tells you the average number of customers at the

checkout counter are 24 customers per hour. The service rate by the checkout

worker averages 30 customers per hour. You are trying to justify whether to hire

another sales associate to corporate headquarters.

Problem 2.3 (Continued)

You need to determine the following statistics for

one and two cashiers:

1. The expected number of customers in the queue

2. The expected number of customers in the system

3. The expected time waiting in line at the register

4. The expected time waiting in line plus the time being served

5. The mean service time for a customer

6. The probability of having 5 customers in the system

7. The proportion of time the one cashier can be expected to be busy

Problem 2.3

2 21. / ( ) (24) / 30(30 24) 3.2

2. /( ) 24 /(30 24) 4

3. /( ( )) 24 /(30(30 24)) 24 / 180 .133 . 8m inutes.

4. 1 /( ) 1 / 6 .167 . 10

5.1 / 1 / 30 / .033 /

q

s

q

L Customers

L Customers

W hrs

W s hrs m inutes

customers hr hrs c

5 5

2 /

6. (0) 1 / 1 1 (24 / 30) .2, (5) (0) .2(.8) .066

7. / 24 / 30 .8 exp

usstomer m inutes customer

P P P

proportion of time cashier ected to be busy

Problem 2.3 (Continued)

For 2 Cashiers (S=2)

* See attached excel spreadsheet for P(0), P(5) Calculations

*2

2 2

1. (0)( / ) / !(1 ) , (0) .4286

(.4286)(24 / 30) (24 / 2 * 30) / 2 !(1 24 / 2 * 30) .152

2. ( / ) .152 (24 / 30) .952

3. / .152 / 24 / .0063

s

q

q

s q

q q

L P s P from mltiserver table

L customers

L L Customers

W L customers customers hour

5 ( 5 2 )

3 .381

4. (1 / ) .381 m in/ 2 m in/ 2.381 m in/

5.1 / ( )

6. (5) (24 / 30) / 2 !2 * .4286, (0) .4286

7. / 24 / 2 * 30 .40

q

hrs m inutes

W s W customer customer customer

mean service time for a customer

P P

s

Using Formulas on Page 2-15 Appendix 2B

P(5) =.008777

Some Insights Into Single Server Analysis

What is a best measure of how well a system a

system is working?

= server utilization

Ws = time in system

Wq = time waiting in line

To me, from a customer service perspective, Wq is

the most important measure!

Homework Problem 2.4

Hoboken police have 10 patrol cars. A patrol car breaks down and requires service once every 15 days. The police department has 2 repair workers each of whom takes an average of 3 days to repair a car. Breakdown and service times are EXPONENTIALLY distributed.

Limited Source (k=10), s=2, = 1/15 = .0667 cars/day, =.333 cars/day, N = calling population =10

1. Determine avg. number of cars in good condition

number of good cars = N Ls so we need to find Ls (limited source and s=2)

P(1)=.2404, P(2)= .2163, P(3) = .1731, P(4)=.1211, P(5)=.0727, P(6)=.0363, P(7)=.0145

P(8)=.0044, P(9)=.0009, P(10)=.0001 Ls=2.4037, N-Ls=7.5963 Cars expected to be good!

2. Determine the average downtime for a police car that needs repairs.

3. Determine the fraction of time at least one worker is idle.

1. P(0)+P(1)=.1202+.2404=.3606

# 1 (1) 2 (2) ..... 10 (10)s

L Expected inSystem P P P

( ) .0667(7.5963) .5064 /

/ 2.4037 / .5064 4.7465 sin

k s

s s k

See finit source eq table

N L cars day

W L day system

=.1202

for n s PN

N n nfor s n N P

N

N n s s

n

n s

n

0 0 0

( )

!

( ) ! !( )

!

( ) ! !

1

0 2

(0) 1 /[ !( / ) /( ) ! ! !( / ) /( ) ! ! ]n s

s N

n n

n n

P N N n n N N n s s

See Formulas in Table 2.3 on Page 2-7

Problem 2.4 Spreadsheet

Lambda Mu S N K RHO

0.06666667 0.333333333 2 10 10 0.2

X Y X+Y P(0)

3 5.320431 8.320431 0.1202

P(0) 0.1202

P(1) 0.2404

P(2) 0.2164

P(3) 0.1731

P(4) 0.1212

P(5) 0.0727

P(6) 0.0363

P(7) 0.0145

P(8) 0.0044

P(9) 0.0003

P(10) 0.0001

0.999592

Lets look at the Excel Spreadsheet

GPSS model of Problem 2.4

Repair Storage 2

XPDIS Function RN1,C24

0,0/.1,.104/.2,.222/.3,.355/.4,.509/.5,.69/.6,.915/.7,1.2/.75,1.38

.8,1.6/.84,1.83/.88,2.12/.9,2.3/.92,2.52/.94,2.81/.95,2.99/.96,3.2

.97,3.5/.98,3.9/.99,4.6/.995,5.3/.998,6.2/.999,7/.9998,8

Generate 360,FN$XPDIS,,10 ; 15 days X 24 Hours per day

Advance 0

Queue1 Queue Broken

Queue Repair

Enter Repair

Depart Repair

Advance 72,FN$XPDIS ; 3 days X 24 Hours Per Day

Leave Repair

Depart Broken

Queue Good

Advance 360,FN$XPDIS ; 15 days X 24 Hours Per day

Depart Good

Transfer ,Queue1

Generate 24 ; 1 Day X 24 Hours Per Dy

Terminate 1

GPSS Model of Problem 2.4 Results

QUEUE MAX CONT. ENTRY ENTRY(0) AVE.CONT. AVE.TIME AVE.(-0) R

REPAIR 8 0 505566 224900 0.879 41.728 75.165 0

BROKEN 10 2 505566 0 2.396 113.734 113.734 0

GOOD 10 8 505564 0 7.603 360.951 360.951 0

STORAGE CAP. REM. MIN. MAX. ENTRIES AVL. AVE.C. UTIL. RETRY DELAY

REPAIR 2 0 0 2 505566 1 1.517 0.758 0 0

Summary

One lecture overview of queuing some universities teach whole classes in queuing

Understanding of queuing necessary for simulation

These are closed form math solutions Modeling queues with simulations is a better technique than closed form if the data is available

Questions?

Process Generators

Continuous Process Generators

In many simulations, it is more

realistic and practical to use

continuous random variables

- Computational it is more efficient

- More representative of the

real world

Discrete Process Generators

Easier to explain and

understand

Another DPG Example

Suppose a random variable can take on the values 2, 5, 8, 9, 12 with corresponding relative frequencies .15, .20, .25, .22, and .18.

DPG:

Value of Relative Cumulative r1 < r

Corresponding GPSS Function

CLASS FUNCTION RN4,D5

.15,2/.35,5/.6,8/.82,9/1,12

Note: GPSS RN4 can result in a 0 but not 1.0 so not quite table in

previous slide but not much of a concern.

Discrete Process Generators

0

2

4

6

8

10

12

Frequency

b. Histogram

0 0.5 1.0 0.25 0

0.25

0.50

0.75

1

a. Sorted Observations

12

9

3

6

TimeBetween

TruckArrivals,Hours

0

0.4

0.8

1.0

CumulativeProbability

c. Cumulative Distribution Function

0 0.5 1.0 0.25

0.75

0.75

0.2

0.6

Uniform Time Between Random Truck ArrivalsVariable (hours)

0 < r < 0.40 0.250.40 < r < 0.70 0.500.70 < r < 0.80 0.750.80 < r < 1.00 1.00

d. Process Generator

Time Between Truck Arrivals, Hours

12/30

21/3024/30

30/30

Time Between Truck Arrivals, Hours

10 20 30

Role of Statistics in Modeling and

Simulation

Frequency tables or histograms suppress much detail

Idealized mathematical representations are needed

real world is not discrete

mathematically efficient

The curve describing the shape of the distribution is called

a frequency curve

0

0.1

0.2

0.3

0.4

5 6 7 8 9

Interarrival Time, Students/Hour

f(x)

Area Under

Curve Must Be

Equal to 1

GPSS Basic Single Server Model

Lets RUN a GPSS model of a Basic Single Server augmented to obtain histograms of inter arrival time distribution, number of

arrivals per unit time distribution, and waiting time distribution.

Lets look at EXCEL plots of the distribution

Are you convinced that math and simulation model agree?

Basic Single Server w/ Histograms

XPDIS Function RN1,C24

0,0/.1,.104/.2,.222/.3,.355/.4,.509/.5,.69/.6,.915/.7,1.2/.75,1.38

.8,1.6/.84,1.83/.88,2.12/.9,2.3/.92,2.52/.94,2.81/.95,2.99/.96,3.2

.97,3.5/.98,3.9/.99,4.6/.995,5.3/.998,6.2/.999,7/.9998,8

IAV Variable C1-X1

ARR Variable X2

HSTIA TABLE V$IAV,0,100,20

ARRT TABLE V$ARR,0,1,20

INQUE QTABLE 1,0,100,30

Generate 360,FN$XPDIS

PATH Tabulate HSTIA

Savevalue 1,C1

Savevalue 2+,1

Queue 1

Seize 1

Advance 240,FN$XPDIS

Release 1

Depart 1

Terminate 1

Generate 1440

Tabulate ARRT

Savevalue 2,0

Terminate

Plots the Arrival and Wait Time Distributions (What arrival

distribution do you think we will get?)

Exponential Inter Arrival and Service Times

Interarrival Time Distribution

Interarrival Distribution IA

0

5000

10000

15000

20000

25000

30000

0 2 4 6 8 10 12 14 16 18 20

Interarrival Time

Fre

qu

en

cy

Series1

Minutes x 100

Arrival Distribution

Arrivals Rate - Poisson

0

1000

2000

3000

4000

5000

6000

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Arrivals Per Day

Fre

qu

en

cy

Series1

Time Spent in Queue Distribution

Time in Queue Distribution

0

2000

4000

6000

8000

10000

12000

14000

0 10 20 30

Minutes x 100

Fre

qu

en

cy

Series1

Generating Exponential IAs Using EXCEL

Need to Convert Uniform Random Numbers (RAND()) to values of the Random Variable with desired Exponential Distribution

We could approximate using a Discrete Process Generator but better to use a Continuous Process Generator when dealing with continuous

distributions such as the exponential.

We will learn a method called the Inverse Transform Method to derive a formula which can yield the desired distribution. For the exponential

that will be:

(1 / ) ln( )

0 1

x r

where r

arrival rate

(1 / ) * ln( ())X RAND

Lets Checkout using Excel!

Next Week

Read Chapter 4

Examples of Simulation

Class Problems 4. 2, 4.3, 4.7(Ginos)*

Role of Probability & Statistics

Continuous Process Generators

*see attached word doc for CORRECT Ch 4 problems to do!!!!