l-3 simulation of mm1

8/2/2019 L-3 Simulation of MM1

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Computer-based SimulationLect-3

Simulation of a single-server queuing system

International University for Science and Technology

Faculty of Information Technology

Department of Software Engineering

Muhammad WANNOUS, PhD


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Outline

Problem statement

Determining events and variables

Statistical counters

Detailed explanation

Simulation output and discussion


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Problem statement

Consider a single-server queuing systemfor which the inter-arrival times A1,A2,are Independent and IdenticallyDistributed (IID) random variables (have

the same probability distribution). A customer who arrives and finds the

server idle enters service immediately, andthe service times S1, S2,of the

successive customers are IID randomvariables that are independent of the inter-arrival times.


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Problem statement

A customer who arrives and finds the server busyjoins the end of a single queue. Upon completingservice for a customer, the server chooses acustomer from the queue (if any) in a FIFO manner.

The simulation begins in the empty and idle state(no customers in queue) in t=0 and the server waitsfor the arrival of the first customer which occurs at t0+ A1.

We wish to simulate this system until a fixed numberof customers (n) have completed their delays in thequeue.


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Problem statement

To measure the performance of this system, we will look atthe estimates of 3 factors:

The average delay in the queue of the n customers d(n)

The average number of customers in the queue q(n)

The utilization of the server u(n)

For simplicity, consider the following values:

A1 = 0.4

A2 = 1.2

A3 = 0.5

A4 = 1.7A5 = 0.2

A6 = 1.6

A7 = 0.2

A8 = 1.4

A9 = 1.9

Mean = 1

S1 = 2.0

S2 = 0.7

S3 = 0.2

S4 = 1.1S5 = 3.7

S6 = 0.6

Mean = 0.5


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There is no completely general way todetermine the number and definition ofevents in general for a model.

It is also difficult to specify the state variables,especially in complex models.

There are some principles and techniques to

help simplify the structure and avoid logicalerrors.


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The event-graphmethod.

Events are represented by nodes.

The nodes are connected by directed arrowsshowinghow events can be scheduled from other events/themselves.

One thick arrow indicates that one event at the end ofthe arrow maybe scheduled from the beginning of thearrow in a (possibly) nonzero amount of time (delay).

One thin arrow indicates that one event at the end ofthe arrow maybe scheduled from the beginning of thearrow immediately.


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Arrival Departure

Arrival DepartureEnterService


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Strongly-connectedcomponents are the node(s) within which it is possible to move from

every node to every other node by followingthe arrows in their directions.

In any strongly-connected component ofnodes that has no incoming arrows from otherevent nodes outside the component theremust be at least one node that is initiallyscheduled.


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Arrival Departure

Arrival DepartureEnterService


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For simplicity the event node that hasincoming arrows that are thin can be

eliminated (enter service node is eliminated). Endof-simulation can be introduced as an

event node.


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Arrival Departure

End ofsimulation


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For each customer ithe delay is denoted by Di.

The average delay of n customers is then can becalculated as:

estimated

D

n

Di

n

i

n

i

d

^

0

1

^


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The number of customersjin the queue is donatedby Q(t) and this number of customers continued fora period Tj.

areaTtQ

T

TtQ

n

T

jT

n

jj

n

j

jj

n

jj

q

q

0

^

0^


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Server utilization u(n) is the time when the serverwas busy Tk compared to the total time of simulationT(n)

areaBT

busyidleB

T

BTn

k

n

k

k

u

1

0

0

^


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0

0.5

1

1.5

2

2.5

3

3.5

0

0.

3

0.

6

0.

9

1.

2

1.

5

1.

7 2

2.

2

2.

4

2.

7 3

3.

2

3.

5

3.

8 4

4.

3

4.

6

4.

9

5.

1

5.

4

5.

6

5.

8

6.

1

6.

4

6.

7 7

7.

2

7.

5

7.

8

8.

1

8.

4

8.

6

8.

9

t

Q(t),B(t)


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n= 6 delays in queue desired Hand simulation:

Display system, state variables, clock, event list, statistical counters allafterexecution of each event

Use above lists of inter-arrival, service times to drive simulation Stop when number of delays hits n= 6, compute output performance

measures


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Detailed explanationArrivalevent

Schedule the nextarrival event

Server busy?Add 1 to the

number in queue

Store time of arrivalof this customer

Set delay =0

Add 1 to thecustomers delayed

Set server = busy

Schedule departureevent for this

customerReturn

YesNo


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Detailed explanationDeparture

event

Queue empty?Server = idle

Eliminate departureevents from

consideration

Subtract 1 from thenumber in queue

Compute delay forthe customer

entering service

Add 1 to thenumber of

customers delayed

Schedule departure

event for thiscustomer

Return

YesNo

Move eachcustomer in queue

up 1 place


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Interarrival times: 0.4, 1.2, 0.5, 1.7, 0.2, 1.6, 0.2, 1.4, 1.9,

Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6,


Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6,


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Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6,


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Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6,


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Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6,


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Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6,


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Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6,


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Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6,


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Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6,


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Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6,


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Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6,


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Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6,


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Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6,


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Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6,


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Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6,

Final output performance measures:Average delay in queue = 5.7/6 = 0.95 min./cust.

Time-average number in queue = 9.9/8.6 = 1.15 custs.

Server utilization = 7.7/8.6 = 0.90 (dimensionless)


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Program Organization and Logic

C program to do this model (FORTRAN as well is in book)

Event types: 1 for arrival, 2 for departure

Modularize for initialization, timing, events, library, report, main

Changes from hand simulation:

Stopping rule: n= 1000 (rather than 6) Interarrival and service times drawn from an exponential

distribution (mean b= 1 for interarrivals, 0.5 for service times)

Density function

Cumulative distribution function


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Program Organization and Logic(contd.)

How to draw (or generate) an observation(variate) from an exponential distribution?

Proposal: Assume a perfect random-number generatorthat

generates IID variates from a continuous uniformdistribution on [0, 1] denoted the U(0, 1)distribution see Chap. 7

Algorithm:

1. Generate a random number U2. Return X=b ln U


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Questions and comments

Ask before being asked.

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