performance models- representation and analysis methods

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Chapter six

Performance Models- Representation and

Analysis methods



1. Queuing notation• Queue

– banks, – machine shop,

– airline reservation systems etc

• Optimization of – waiting time, – queue length,

– service to those in queue

• Ideal system – – no queue and – no idle time

• Objective of queuing system- – optimization of queue and wait time



2. Rules for all queues

• Customers arrive at a constant or variable rate

• Customers are to be served at constant or

variable rate



Symbols used

• State of system- number of customers in queuingsystem ( queue and server)

• Queue length – number of customers waiting forservice to begin

• N(t) – number of customers in queuing system at time t• Pn(t)- probability of n customers in queue

• S- number of servers

• n - mean arrival rate of new customers when n

customers are in system• n- mean service rate for overall system when n

customers are in system



Classification of queuing systems

• Queuing systems are classified based on

– Calling source –

• the population from which customers are drawn.

–

The input or arrival process – • distribution of number of arrivals per unit time,

• the number of queues that are permitted to be formed,

• the maximum queue length,

• maximum number of customers desiring service

– The service process –

• time allotted to serve customers,

• number and arrangement of servers,



Assumptions

• Successive arrivals are independent

• Long term inter arrival time constant exist

•

The probability of an arrival taking place intime t is proportional to t



Principles of queuing theory

• Two statistical properties –probability distributionof inter arrival times and probability distributionof service time

• Example: – FIFO service

– Random arrivals• In a given interval of time, only one customer is expected to

come

•Arrival with Poisson distribution

– Steady state

X= Number of arrival per unit timex- number of customers per

unit time

Average arrival per time



Principles of …

• Poisson arrival patter means inter arrival time

is exponential with the same mean



Example: gas filling station

• Car arrival rate 5 minutes between arrival

• Cars arrive according to Poisson process withmean 12cars/hr

• Probability distribution of number of arrivals perhour is

• Distribution of time between two arrivals isexponential



Arrival of K customers at a time

• General Poisson distribution formula

• Where f(t) is given by

• Arrival time – exponential• Number of arrivals – Poisson



Assumptions for service time

• Similar assumptions as of arrival

– Statistical independence of successive servicing

– Long term constant for service time

– Probability of completion is proportional to t

• Exponential service time

Where v is long term average service time



Arrival-service model

• Assumptions used

– Arrival is random

– Arrival from single queue

– FIFO

– Departure is random

– Probability of arrival in t is t

– Probability of departure in t is t



• Probability of being busy

• Average number of customers in servicefacility is

• Probability of no waiting time is (1-)

•

Probability of – 1 customer arriving no customer departing in t

– 1 customer arriving and 1 customer departing in

t

– No customer departing and no customer arriving



Other performance metrics

• Average number of customers at time t

•

Probability of n customers in the system

• Probability of n customers in queue

• Average number of customers in queue



Performance cont…

• Average time a customer spends in system

• Average time a customer spends in queue



Example: customers arrive in bank according to

Poisson process

• Mean inter arrival is 10 minutes

• Average service time in counter 5 minutes

A) what is the probability that customer will not wait

B) what is the expected number of customers in bank

C) how much time is a customer expected to wait in the bank

=6 =12a) b) c)



3. Little’s law

• Is an important tool for verifying queuing

simulations

• Used to determine average number of

customers in system

• It states that

Where

average time a job spends in the system



Stochastic process

• Generating stochastic process is generation of

sequence of variates with probability distribution

• IID process generation

– Generation is similar

– Steps

• 1) determine the seeds for the RNG

• 2) generate random numbers using each seed

• 3) using the CDF of the given distribution, find the inverse

and generate the random sequence elements

• 4) repeat the steps until the required number is obtained



Non IID stochastic process

• Processes should have temporal relations

• Relation is expressed in form of joint

distribution

• Defining temporal relation is difficult



5. Analysis of single queue: birth death

process, M/M/1, M/M/2, M/M/m, M/M/, M/M/m/B

• Kendall’s notation

• V/W/X/Y/Z

– V arrival pattern ( D or M for deterministic or

exponential respectively)

– W service pattern

– X number of servers (take infinity if not specified )

– Y system capacity

– Z queue discipline(FIFO, LIFO)



Simulation of M/M/1/

• Simulate an M/M/1/ system with mean arrival rate of 10

per hour and the mean service rate as 15 per hour for asimulation run of 3 hour. Determine the average customer

waiting time, percentage idle time of the server, maximum

length of the queue and average length of queue

• Average customer waiting time

• Average length of queue

min12hr 2.01015

11s

3

4

15

10

5

10ALQ



Result obtained using simulation

program

• Exponential arrival and service time is used

– r=rand()/32768

– Iat=(-1./mue)*log(1-r)

• Time is counted in minutes• For a single run

– Number of arrivals=50

– Average waiting time=30.83minutes – Average server idle time=2.46

– Maximum queue length=16



Single queue multiple servers

M/M/s/

• Let

– s denote number of servers in system

– Each server provides service at the same rate

– Average arrival rate for all n customers is same

– <s



M/M/s/ analysis cont…

• For n busy servers, the over all service rate is

n

• Probability that there are (n+1) customers is

given by( n>s)

• Where

• (n-s) customers are waiting



M/M/s/ analysis cont…

• Probability that all servers are busy is

– probability that n s

– This is given by



M/M/s/ cont…

• Average length of queue

=



Example – 2 server M/M/2/

• In a service station with two servers,

customers arrive at an average rate of 10 per

hour. The service rate of each server is 6

customers/hour.

• Determine

– A) the fraction of time that all servers are busy

– B) average number of customers waiting

– C) average waiting time



• Soln.

• Servers will be busy if there are n>2 customers

• P(n2 )is then

• where Po is

• Then P(n 2)

2s

hr /6

hr /10

= = 0.79



Analysis using simulation M/M/2/3

• 2 servers

• Maximum capacity of 3

• Exponential arrival and service time

0 9 13 22 26 33

10 7 12 20 15 15

arrival Server 1 Server 2

cust idle service wait idle service wait

0 0 10 0

9 - - - 9 7 0

13 3 12 0 - - -

22 - - - 6 20 0

26 1 15 0 - - -



Exercise

• Write a simulation program to analyze an

M/D/2/3 system

– Exponential arrival with mean 3 minutes

– 2 servers and maximum capacity of 3

– Service time is deterministic with 5 and 7 minute

service time respectively

– Simulate system for 1 hour and determine• Idle time of servers

• Waiting time of customers

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