Queueingmodel

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Queuing Model

2004. 5. 29

M S Prasad


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Queueing Systems

Queue

a line of waitingcustomers who require

service from one or moreservice providers.

Queueing system waiting room + customers

+ service provider

Arrivals

Customers

Queue Server(s) Departures


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Queueing Models

Widely used to estimate desired performance measuresof the system

Provide rough estimate of a performance measure

Typical measures Server utilization

Length of waiting lines

Delays of customers

Applications

Determine the minimum number of servers needed at aservice center Detection of performance bottleneck or congestion Evaluate alternative system designs


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Kendall Notation

A/S/m/B/K/SD

A: arrival process

S: service time distribution

m: number of servers

B: number of buffers(system capacity)

K: population size

SD: service discipline


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Arrival Process

Jobs/customer arrival pattern

form a sequence of Independent andIdentically Distributed(IID) random variables

Arrival times : t1, t2, , tj Interarrival times : j=tj-tj-1

Arrival models Exponential + IID (Poisson)

Erlang Hyper-exponential

General : results valid for all distributions


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Service Time Distribution

Time each user spends at the terminal

IID

Distribution model Exponential

Erlang

Hyper-exponential

General

cf. Jobs = customers

Device = service center = queue

Buffer = waiting position


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Number of Servers

Number of servers available

Single Server Queue

Multiple Server Queue


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Service Disciplines

First-come-first-served(FCFS)

Last-come-first-served(LCFS)

Shortest processing time first(SPT)

Shortest remaining processing time first(SRPT) Shortest expected processing time first(SEPT)

Shortest expected remaining processing timefirst(SERPT)

Biggest-in-first-served(BIFS)

Loudest-voice-first-served(LVFS)


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Common Distributions

M : Exponential

Ek : Erlang with parameter k

Hk : Hyperexponential with parameterk(mixture of k exponentials)

D : Deterministic(constant)

G : General(all)f(t)1/m

tm

f(t)

tm

s

Exponential

General


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Example

M/M/3/20/1500/FCFS

Time between successive arrivals is exponentiallydistributed

Service times are exponentially distributed

Three servers

20 buffers = 3 service + 17 waiting

After 20, all arriving jobs are lost

Total of 1500 jobs that can be serviced Service discipline is first-come-first-served


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Default

Infinite buffer capacity

Infinite population size

FCFS service discipline Example

G/G/1 G/G/1/


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Key Variables

: interarrival time

: mean arrival rate = 1/E[] s : service time per job

: mean service rate per server = 1/E[s] n : number of jobs in the system(queue length) = nq+ns

nq : number of jobs waiting

ns : number of jobs receiving service

r: response time time waiting + time receiving service

w: waiting time

Time between arrival and beginning of service

Service

ArrivalRate

(Average Number

in Queue (Nq )

Average Wait

in Queue (w)

Rate ( Departure


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Littles Law

Waiting facility of a service center

Mean number in the queue

= arrival rate X mean waiting time Mean number in service

= arrival rate X mean service time


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Example

A monitor on a disk server showed that the average

time to satisfy an I/O request was 100msecs. The I/O

rate was about 100 request per second. What was

the mean number of request at the disk server? Mean number in the disk server

= arrival rate X response time

= (100 request/sec) X (0.1 seconds)

= 10 requests


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Stochastic Processes

Process : function of time

Stochastic process

process with random events that can be described bya probability distribution function

A queuing system is characterized by threeelements:

A stochastic input process

A stochastic service mechanism or process

A queuing discipline


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Types of Stochastic Process

Discrete or continuous state process

Markov processes

Birth-death processes Poisson processes

Markov process

Birth-death process

Poisson process


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Discrete/Continuous State Processes

Discrete = finite or countable

Discrete state process

Number of jobs in a system n(t) = 0,1,2,

Continuous state process

Waiting time w(t)

Stochastic chain : discrete state stochasticprocess


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Markov Processes

Future states are independent of the past

Markov chain : discrete state Markov process

Not necessary to know how log the processhas been in the current state

State time : memoryless(exponential) distribution

M/M/m queues can be modeled using Markovprocesses

The time spent by a job in such a queue is aMarkov process and the number of jobs in thequeue is a Markov chain


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......

...

...

121110

020100

PPP

PPP

P

The transition probability matrix

-1 1

2/2

1

2/2

1-2/2

2/2

0 1


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Birth-Death Processes

The discrete space Markov processes in whichthe transitions are restricted to neighboringstates

Process in state n can change only to staten+1 or n-1

Example

The number of jobs in a queue with a single server

and individual arrivals(not bulk arrivals)


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Poisson Processes

Interarrival time s = IID + exponential

Birth death process that k = , k = 0 for all k

Probability of seeing n arrivals in a period from 0 to t

Pdf of interarrival time

nep n

)(

t : interval 0 to t

n : total number of arrivals in the interval

0 to t : total average arrival rate in arrivals/sec


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M/M/1 Queue

The most commonly used type of queue

Used to model single processor systems or individualdevices in a computer system

Assumption Interarrival rate of exponentially distributed Service rate of exponentially distributed Single server

FCFS

Unlimited queue lengths allowed

Infinite number of customers

Need to know only the mean arrival rate() and themean service rate

State = number of jobs in the system


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M/M/1 Operating Characteristics

Utilization(fraction of time server is busy) = /

Average waiting times W = 1/( - ) Wq = /( - ) = W

Average number waiting

L = /( - ) Lq = /( - ) = L


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Flexibility/Utilization Trade-off

Utilization = 1.0= 0.0

LLqWWq

High utilization

Low ops costsLow flexibilityPoor service

LowutilizationHigh ops costsHigh flexibilityGood service

Must trade off benefits of high utilization levelswith benefits of flexibility and service


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Cost Trade-offs

1.0= 0.0

CostCombined

Costs

Cost ofWaiting

Cost ofService

Sweet SpotMin Combined

Costs

*


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M/M/1 Example

On a network gateway, measurements showthat the packets arrive at a mean rate of 125packets per seconds(pps) and the gateway

takes about two milliseconds to forward them.Using an M/M/1 model, analyze the gateway.What is the probability of buffer overflow if thegateway had only 13 buffers? How many

buffers do we need to keep packet loss belowone packet per million?


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Arrival rate = 125pps Service rate = 1/.002 = 500 pps Gateway utilization = / = 0.25 Probability of n packets in the gateway

(1- ) n = 0.75(0.25)n

Mean number of packets in the gateway

/(1-) = 0.25/0.75 = 0.33

Mean time spent in the gateway (1/ )/(1- ) = (1/500)/(1-0.25) = 2.66 milliseconds

Probability of buffer overflow

P(more than 13 packets in gateway) = 13 = 0.2313=1.49 X 10-8 15 packets per billion packets

To limit the probability of loss to less than 10-6 n < 10-6

n > log(10-6)/log(0.25) = 9.96

Need about 10 buffers


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References

The art of computer systems performanceanalysis. Raj Jain

Queuing Theory : Dr. N k Jaiswal

Queueingmodel

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