Queuing SystemsTime-based versus event-based simulation

• Queuing systems as examples of discrete event systems

• The arrival process

• Performance measures

• Steady state

• Queuing theory: Birth and death processes

Time-Based Simulation• In time-based simulation the clock is incremented

in steps of equal length. – We look at the system every minute, day, week, month,

etc.

• For a time-based simulation we do not need to know what happens within a single time step. All variables are updated at the end of the time step.– Typical example: inventory control (open spreadsheet

Time Based Simulation.xls)

Event Based Simulation• Often systems change in a series of jumps (events)

whose time of occurance is random– Examples: arrival or departure of customer / birth or

death

• Modelling assumption: – What happens between events is not important for the

control of the system

• A system whose evolution is driven by the occurance of events at random points in time is called a discrete event system

Event Based Control• The natural control mechanism for a discrete event

simulation is event based: – For every class of events (arrival, departure, etc.) generate the

next occurance time (possibly if an event (e.g. departure) cannot occur before another event (arrival) has taken place)

– Advance the clock to the time of the next event

– Update the system variables and list of next event times

– Repeat until the simulation clock reaches a preset time

• Flow charts are very helpful in setting up event based simulation programs

Entity Based Control• Entity based control:

– An entity is a “player” who is processed through the system (e.g. a customer, an order)

– When the entity enters the system, all necessary variables (e.g. arrival time, service time, arrival of next customer) are assigned to the entity using variables of past entities and RNGs

– Control the number of entitites that pass through the system

• Entity based control is particularly suitable for spreadsheet simulations– Open spreadsheet Event Based Simulation.xls

Queuing Systems

• Time-based versus event-based simulationQueuing systems as examples of discrete

event systems

• The arrival process

• Performance measures

• Steady state

• Queuing theory: Birth and death processes

Queuing Systems

• Events: – Arrivals and departures of customers

– Customers balking or reneging

– Customers moving from one queue to the next in a queuing network

– etc.

• Simple queuing system: – Customers arrive, queue up, are serviced, and depart.

• Queuing networks:– Customers pass through a network of queues

The Queuing Paradigm

• Characteristics of general queuing systems: – Entities flow through a system from one processing point to another – Some element of randomness

• Main sources of randomness: – Arrival times– Specification of arriving entities– Processing times

• Want to optimise the design and control of the system (e.g. manufacturing process)– Design variables, e.g., size, number and layout of processing points – Control variables, e.g., routing of entities through the system

Characteristics of simple queuing systems

• The most important characteristics of a simple queuing system are the distributions of – interarrival times (times between arrivals of two

consecutive customers)

– service times (time for service of single customers).

• It is often assumed that these random variables are (statistically) independent

• Further important characteristics are the number of servers and the system capacity

Kendall’s notation• Simple queuing systems are conventionally labelled by

U / V / s / / W

• U and V denote the interarrival and service time distribution – most important abbreviations are D for deterministic, M for exponential

and G for general (i.e. unspecified) distribution

• s is the number of servers

• is the system capacity (max. # customers in service and queue)

• W is the queuing discipline (e.g. FIFO,LIFO,Priority based)

• and W are optional with default values and W = FIFO

Kendall’s notation (continued)

• Example: an M/M/s queue is a simple queuing system with exponential interarrival and service times, s servers, unlimited system capacity, and FIFO queuing discipline.

• Tacit assumption when Kendall’s notation is used:– interarrival times are independent and identically distributed

(i.i.d.)

– service times are i.i.d.

– interarrival and service times are independent

Uncertainty drivers• Main uncertainty drivers are interarrival times (times

between the arrival of customers) and service times (times for service of a customer)

• They control the evolution of the number N(t) of customers in the system at time t.

• N(t) is called the state of the system at time t

• Further uncertainties could be driving your model (e.g. amount of money spent by customers)

• Frequent assumption: all random variables involved in the model are (statistically) independent

Arrival and service rates

• The arrival rate is the expected number of arrivals per unit time (e.g. per hour)

• The service rate is the expected number of customers that can be served by one of the servers per unit time

• The expected service time per customer is 1/ • The expected time between two arrivals is 1/• Arrival rate and service rate often depend on the time

of the day, week, year,etc. (seasonal rates)

The utilization factor

• Suppose a G/G/s queue– arrival rate – service rate,

• The utilization factor (or traffic intensity)is defined by

s• Interpretation: is the fraction of time we expect

the service facility to be busy (i.e. at least one of the servers to be busy)

• If then the queue “explodes”• What if

An average case argument for

• Suppose s=1 (to make life easy)• If we simulate with

interarrival times = average interarrival time (1/ service times = average service time (1/

then we obtain no queue• Ergo: the average queue length is zero?• Wrong: This is yet another example of the flaw of

the averages• Let’s look at a spreadsheet model (open Waiting Time

Versus Service Variance.xls)

Queuing Systems

• Time-based versus event-based simulation

• Queuing systems as examples of discrete event systems

The arrival process

• Performance measures

• Steady state

• Queuing theory: Birth and death processes

The Role of the Exponential Distribution

• What is a reasonable distribution for interarrival times? • What are typical properties of interarrival times? • An empirical observation: You have been sitting on a lake for half an hour trying to catch

a fish - without success. Then your friend who is half an hour late as usual, joins you. Although you have already been sitting there for half an hour, your friend’s chances of catching a fish in the next 10 minutes are the same as yours.

The Lack-of-Memory Property

• The empirical observation can be formalized as

P(X>t+s ¦ X>s)=P(Y>t),

where X and Y are the time that elapses until you and your friend, resp., catch a fish.

• Since both waiting times have the same distribution, i.e. P(X>t)=P(Y>t), we obtain

P(X>t+s ¦ X>s)=P(X>t).

• This property of a random variable is called the lack-of-memory property

“Probability of X>t+s, given X>s” (Conditional Probability)

Interarrival Time Distributions• The lack-of-memory property is typical for

interarrival times, provided arrival rates are constant over time

• Important Mathematical result: If a continuous random variable (with a continuous density function) has the lack-of-memory property then it is exponentially distributed

• This mathematical result is the reason for the prevalence of the exponential distribution in queuing models

The exponential distribution• Parameter • Density function

f(x)=e-x for x>0 and f(x)=0 for x• Cumulative distribution function

F(x)=1-e-x for x>0 and F(x)=0 for x• Mean 1/ Standard deviation 1/• Conclusion: If interarrival times have the lack-of-

memory property then they are exponentially distributed with parameter arrival rate.

Spreadsheets

We have seen before that, by using the inverse transform method, an exponentially distributed random variable with parameter can be generated via

=-ln(1-rand())/A1 where cell A1 contains the value

Equivalently: -ln(rand())/A1 because “1-rand()” is a uniform RNG since rand() is a uniform RNG)

A histogram of 1000 trials with =2

Histogram

050

100150

200

0.1

0.4

0.7 1

1.3

1.6

1.9

2.2

2.5

2.8

Mo

re

Bin

Fre

qu

en

cy

.00%

50.00%

100.00%

150.00%

Frequency

Cumulative %

Service times• The lack-of-memory property is less typical for

service times S.

• However, in some situations the characteristics of the exponential distribution apply to the service time distribution– e.g. P(t St+t) decreases with t for every fixed t>0.

• Drawback of exponential: mean = standard deviation

• Whatever distribution is chosen for service or interarrival time, a goodness of fit test with sampled data should be used to check its validity– If in doubt use re-sampling from historical data

The minimum property

Mathematical result:

If X1,...,Xn are independent exponentially distributed random variables with parameters 1,...,n then

X=min{X1,...,Xn}

is exponentially distributed with parameter

1+...+ n .

Why is this important?• Example: If s independent exponentially

distributed servers with service rates 1,..., s are busy then the time until the next service completion is exponentially distributed with parameter = 1+...+s.

• Hence if the utilization factor is close to 100% (heavy traffic) then the behaviour of an M/M/s system with service rate approaches the behaviour of an M/M/1 system with service rate s

Disaggregation of arrival processes

Mathematical result:

If customers arrive with exponential interarrival times at rate and there are n types of customers with pi being the probability that a customer of type i arrives then the interarrival times of customers of type i are exponentially distributed with parameters i = pi

The Poisson Process• Let A(t) denote the number of customers that

arrive in the time interval [0,t].• Mathematical result: If the interarrival times are

independent and exponentially distributed with parameter then A(t) has a Poisson distribution with parameter t, i.e.

P(A(t)=n)=e-t((t)n/n!) • A(t) is called a Poisson arrival process• E[A(t)]=V[A(t)]= t (mean=variance)

Nonstationary Arrival Processes• One characteristic of a Poisson arrival process is

P(A(t+t)=n+1 A(t)=n) t

where the arrival rate is constant over time.• This is often an unrealistic assumption• A nonstationary (or nonhomogeneous) Poisson process

allows the arrival rate to vary over time:

P(A(t+t)=n+1 ¦ A(t)=n) tt.• The function (t) is called the arrival rate function

Generating nonstationary arrivals• Naïve implementation: Generate at time t arrival with rate

(t)• It’s better to use a thinning process:

– Let max be an upper bound for the arrival rate function, i.e. (t) max for all t

– Suppose the simulation clock has reached time t.• Generate an interarrival time x for the stationary process with

arrival rate max • Increment clock to t+x• Accept arrival with probability (t)/max (i.e. generate a

uniform RV u and neglect arrival if u> (t)/max )• See spreadsheet Nonstationary Arrival Rates.xls

Queuing Systems

• Time-based versus event-based simulation

• Queuing systems as examples of discrete event systems

• The arrival processPerformance measures

• Steady state

• Queuing theory: Birth and death processes

Performance Measures

• Queuing models are often used to improve customer satisfaction

• The main performance measures are queue length and waiting time

• What do we mean by that? – Open spreadsheet Performance Measures.xls

What are we interested in?

• The average queue length seen by the first N customers – is a random variable (changes with F9 key)

– depends on N (moderate changes as N gets large)

• Mathematically: Interested in the distribution of the average as N tends to (long run average)

• Practically: Interested in the distribution for large N (simulation)

Average waiting time

• Can simulate waiting times for each customer

• Can simulate average waiting time of first N customers

• Average waiting time of first N customers – is a random variable – depends on N (becomes smoother as N grows)

Queuing Systems

• Time-based versus event-based simulation

• Queuing systems as examples of discrete event systems

• The arrival process

• Performance measuresSteady state

• Queuing theory: Birth and death processes

Focus on simple queuing systems

• Process: Customers arrive -> queue up -> are served (FIFO) -> leave

• Characteristics– Interarrival time– Service time– Number of servers– System capacity

Steady State

• The state N(t) of a queuing system at time t is the number of customers in the system (i.e. in the queue or in service) at time t

• The system is said to be in steady state if P(N(t)=n) does not change with t any more. – Simple queuing system reaches steady state only if <1

• Let’s look at a spreadsheet… (open Steady State.xls)

Approaching Steady State Distribution

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 100 200 300 400 500 600 700 800 900 1000

Customer

% E

arlie

r Cus

tom

ers

obse

rvin

g N(

t)=n

n=0

n=1

n=2

n=3

n=4

n=5

n=6

n=7

n=8

n=9

n>9

Steady State Distribution

• If the system is in steady state then the distribution characterized by the probability mass function pn=P(N=n) is called the steady state distribution – Mathematically, pn is defined to be the limit of P(N(t)=n) as t

tends to infinity (provided the limits exist and the pn add up to 1).

• In simulation experiments it is often advisable to allow for a run-in time before measurements (e.g. waiting times) are taken in order to allow the system to reach steady state.

Average Queue Length

• Expected number of customers in the system (in steady state) is

L=1p1+2p2+3p3+4p4+...

• Expected queue length of an s-server system (in steady state) is

Lq = 1ps+1+2ps+2+3ps+3+4ps+4+...

Average Waiting Time

• W: average waiting time (until service completion) for a customer in the system (in steady state)

• Wq: average waiting time for a customer in the queue

• Relationship: W=Wq+1/

• Little’s formula clarifies the relation between Wq and Lq….

Little’s Formula• In steady state a newly arriving customer expects to see a

queue of length Lq

• She is expected to be in the queue for Wq time units

• During these Wq time units one expects that Wq new customers arrive

• So the queue length is expected to be Wq when the customer leaves the queue

• In steady state the expected queue length when a customer arrives is the same as the expected queue length when she leaves the queue; hence we obtain

Lq = Wq

Queuing Systems

• Time-based versus event-based simulation

• Queuing systems as examples of discrete event systems

• The arrival process

• Performance measures

• Steady stateQueuing theory: Birth and death processes

Queuing Theory Traditional queuing theory is concerned

with obtaining closed form solutions for steady state probabilities

pn=P(N=n)

or the performance measures

L,Lq,W, and Wq

for simple queuing systems.

The Birth and Death Process

• A simple queuing system is a special case of a birth and death process

• “Birth” = “arrival of customer”• “Death” = “departure of customer”• State N(t) is the size of a population at time t• Births and Deaths happen randomly with rates

possibly depending on the size of the population.

Assumptions for B & D Process• Given time t and state N(t)=n, the probability

distribution of the remaining time until the next birth is exponential with parameter n.

• Given time t and state N(t)=n>0, the probability distribution of the remaining time until the next death is exponential with parameter n.

• The random variables of the above assumptions are mutually independent.

• The process is in steady state, i.e. P(N(t)=n) is independent of the time t.

The Balance Equation• At any time the next state transition is either from

n to n+1 or from n to n-1, depending on whether a birth or a death occurs next.

• Define En and Ln to be the rate (average number of events per unit time) at which the system enters and leaves state n, respectively.

• Balance equation:

En = Ln

Rate in = Rate out

Steady State Probabilities

• Recall that pn=P(N=n).

• Notice that pn can be interpreted as the proportion of time the process is in state n

• Idea: Express En and Ln in terms of steady state probabilities pn and use balance equation to calculate pn.

A Formula for E0

• The process can only enter state 0 from state 1

• p1 is the proportion of time the process is in state 1

• is the rate at which the process enters state 0 from state 1

• Hence E0= p1

Further formulas

Similar arguments show that

L0= p0

En=pn-1n-1+pn+1n+1

Ln=pn (n+n)

The Balance Equation Revisited

• Using the balance equation E0=L0 we obtain

p1= (0/1)p0

• Using the balance equations En=Ln for n>0 we further obtain

pn+1= (n/n+1)pn +(/n+1)(npn -n-1pn-1)

• Given p0, we can calculate pn recursively as

pn= cnp0 with cn= (01...n-1(12

...n

Calculating p0

• Notice that

1 = p0+p1+p2+...

= p0+c1p0+c2p0+...• Hence

p0 = 1/(1+c1+c2+...)• In practice one either uses an appropriate formula for the

infinite sum or approximates it by a finite sum. • If the queuing system has finite capacity and

n>then n-1=0 and hence cn=0

Conclusion• The steady state probabilities pn=P(N=n) of a birth

and death process with birth rates n and death rates n are given by

pn = cn/(c0+c1+c2+...) with

c0 = 1,

cn = (01...n-1(12

...nfor n>0.

• Let’s now apply this….

The M/M/1 queue• For the M/M/1 queue we have n= , n= for all n• Hence cn= (nn

and

s=c1 +c2 +...= 1 +2+... = /(1-)• Hence p0 = 1/(1+s) = 1-,

pn= cn p0 = (1- n (geometric distribution)

• Thus L = 1p1+2 p2+3p3+... = (1- (12+33+...) = =

The M/M/1 Queue (continued)

• Given the formula for L we can obtain formulas for the other performance measures W,Wq,Lq by using Little’s formula and the relation W=Wq+1/

• Interesting additional mathematical result: The waiting time in an M/M/1 queuing system (including service) is exponentially distributed with parameter -

The M/M/s Queue A similar (but rather messy) calculation gives the

following formulas for the M/M/s queue (recall that (s))

ps

n

s

s

Ls

sp

n s

n

s

q

s

0

1

2 0

1

11

1

( )

!

( )

!( )

( )

!( )

Finite Calling Population

• Finite calling population: There is a maximum of C customers, i.e. if n customers are in the system then there are only C-n customers remaining in the input source– n=(C-n) for n=0,…,C, n=0 for n>C

• Formulas for steady state probabilities are available

• Ref. Hillier/Lieberman, Introduction to Operations Research, p. 689 ff

Priority Systems• Customers are assumed to belong to m priority

classes (e,g, Class 1 having the highest priority)• Formulas for average waiting times and queue

lengths in the various classes are available, provided interarrival and service times are exponentially distributed

• Ref. Hillier/Lieberman, Introduction to Operations Research, p706 f

The Pollaczek-Khintchine Formula

• The formulas developed so far assumed exponential interarrival and service times

• Formulas for performance measures of more general queuing systems are rare.

• Exception is the M/G/1 queue. If mean E and standard deviation S of the service time are known then the average queue length can be calculated by the Pollaczek-Khintchine formula:

Lq=[S2+E2)]E)• Need E=<1

P-K Formula for Waiting Time

• Little’s formula says Lq=Wq

• Hence

Wq=[S2+E2)]E)

• Affine-Linear Function of Variance S2

– Compare to simulation results in Waiting Time Versus Service Variance.xls

Why Queuing Theory?

• If the assumptions apply then we obtain nice closed form solutions for performance measures

• The formulas give us important indications about the nature of the dependence of performance measures on design parameters – Example: Average queue length in an M/G/1

model grows quadratically with the standard deviation of the service times.

Queuing Theory and Simulation

• By specifying our simulation model so that it obeys the assumptions of the theory we are able to validate the model by comparing its output with the theoretical results

• Example: The simulation model of a single server queuing system with nonstationary arrival rates can be validated by choosing constant arrival rates and then checking the results against the P-K-Formula

Behavioural Phenomena• A realistic simulation of a queuing system has to take the

behaviour of the customers during the queuing time into account

• Typical phenomena are

– Customers do not join the queue upon arrival (balking)

– Customers leave the queue before being served (reneging)

– Customers switch from one queue to another (jockeying)

• See spreadsheet Balking Reneging.xls

Key learning points about Queuing Systems

• Simple queuing systems can be simulated in a spreadsheet

• The prevalence of the Poisson arrival process in queuing models is due to the lack-of-memory property

• Main performance measures are long-run average queue length and waiting times

• Little’s formula allows us to calculate all performance measures L,Lq,W,Wq, once one of them is known

• The balance equation of the birth-and-death process allows us to find closed form solutions for the performance measures of simple queuing systems

Homework

Consider an M/M/1 versus and M/D/1 queue– which one do you expect to have longer waiting

times and why? – Using the template provided on the course

homepage, verify Little’s formula experimentally for both queuing systems by estimating L and W through simulation.

Homework • Compare the theoretical results for the waiting time in an

M/M/1 queuing system with your former simulation results of an M/M/1 system for various run lengths (50,150,300,500 customers)

• Implement formulas for an M/M/1 queue with maximal system capacity in a spreadsheet. (i.e. given the parameters the spreadsheet calculates L,Lq,W,Wq). Does Little’s formula hold?

• Argue that in an M/G/1 queue the average queue length grows linearly with the variance of the service time. What is the rate of growth? Check this by simulation (use the spreadsheet model Waiting time versus Service Variance.xls)