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8/12/2019 Performance Models- Representation and Analysis Methods
http://slidepdf.com/reader/full/performance-models-representation-and-analysis-methods 1/31
Chapter six
Performance Models- Representation and
Analysis methods
8/12/2019 Performance Models- Representation and Analysis Methods
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8/12/2019 Performance Models- Representation and Analysis Methods
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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
8/12/2019 Performance Models- Representation and Analysis Methods
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2. Rules for all queues
• Customers arrive at a constant or variable rate
• Customers are to be served at constant or
variable rate
8/12/2019 Performance Models- Representation and Analysis Methods
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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
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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,
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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
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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
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Principles of …
• Poisson arrival patter means inter arrival time
is exponential with the same mean
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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
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Arrival of K customers at a time
• General Poisson distribution formula
• Where f(t) is given by
• Arrival time – exponential• Number of arrivals – Poisson
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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
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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
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• 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
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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
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Performance cont…
• Average time a customer spends in system
• Average time a customer spends in queue
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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)
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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
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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
8/12/2019 Performance Models- Representation and Analysis Methods
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Non IID stochastic process
• Processes should have temporal relations
• Relation is expressed in form of joint
distribution
• Defining temporal relation is difficult
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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)
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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
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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
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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
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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
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M/M/s/ analysis cont…
• Probability that all servers are busy is
– probability that n s
– This is given by
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M/M/s/ cont…
• Average length of queue
=
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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
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• 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
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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 - - -
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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