Pavel Shaydo > Simulation-DiscreteEvent-0.09 > Simulation::DiscreteEvent::Cookbook::MM10

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NAME ^

Simulation::DiscreteEvent::Cookbook::MM10 - Modelling M/M/1/0 queue

MODELLING M/M/1/0 QUEUE ^

A Bit of Theory

M/M/1/0 queue is usually first example in any book on queueing theory. Let's see how we can build the model of this system using Simulation::DiscreteEvent.

In M/M/1/0 system customers arrive in random moments of time independently of each over. The interval between moments of arrival is a random variable distributed exponentially, i.e. its density function is f(t)=lambda*exp(-lambda*t), t>0. The lambda parameter is arrival rate - the average number of customers arriving in a unit of time. There's a single server in the system which serves customers. The time required to serve a customer is exponentially distributed random variable. The density function is f(t)=mu*exp(-mu*t), t>0. The mu parameter determines the average number of customers that can be served in a unit of time. If customer arrives while server is still busy with another customer, then new customer is turned away and leaves the system. Let's try to determine probability that arrived customer will be turned away.

In discrete-event simulation we assume that system may change its state only in discrete moments of time then some event occurs. Between events the state of the system is constant. Before simulation we're scheduling some events for certain moments, and then during simulation handling these events. Inside event handler we can schedule some additional events, so simulation may run endlessly. We can limit number of events that will be generated, or specify model time at which simulation should be stopped.

Building a Model

The base class for model is Simulation::DiscreteEvent. Instance of this class manages events' queue and invokes event handlers. Here's how we will use it:

    use Simulation::DiscreteEvent;
    my $model = Simulation::DiscreteEvent->new();

Creating Server Class

Now we should add server unit to the model. Server unit contains event handlers. Each server is inherited from Simulation::DiscreteEvent::Server class. This class contains dispatcher that invokes different handlers depending on event, so our server class should only define event handlers and specify for each handler on which event it should be invoked. Here's the server class:

    package Simulation::DiscreteEvent::CB::MM10;
    use Moose;
    use Math::Random qw(random_exponential);
    BEGIN { extends 'Simulation::DiscreteEvent::Server' }

    # server state
    has busy => ( is => 'rw', default => 0 );

    # arrival rate
    has lambda => ( is => 'rw', required => 1 );

    # serving rate
    has mu => ( is => 'rw', required => 1 );

    # number of served customers
    has served => (
        is      => 'rw',
        traits  => ['Counter'],
        default => 0,
        handles => { inc_served => 'inc' }
    );

    # number of rejected customers
    has rejected => (
        is      => 'rw',
        traits  => ['Counter'],
        default => 0,
        handles => { inc_rejected => 'inc' }
    );

    # New customer arrived
    sub arrival : Event {
        my $self = shift;
        my $next_time = $self->model->time 
            + random_exponential( 1, 1 / $self->lambda );
        $self->model->schedule( $next_time, $self, 'arrival' );
        if ( $self->busy ) {
            $self->inc_rejected;
        }
        else {
            my $srv_time = $self->model->time
                + random_exponential( 1, 1 / $self->mu );
            $self->model->schedule( $srv_time, $self, 'finished' );
            $self->busy(1);
        }
    }

    # Customer served
    sub finish : Event(finished) {
        my $self = shift;
        $self->inc_served;
        $self->busy(0);
    }

    no Moose;
    __PACKAGE__->meta->make_immutable;

As you can see we're using Moose, if you don't know what Moose is, you should read Moose::Manual. We're defining several attributes and two event handlers. Each event handler is marked with Event method attribute, e.g. finish method has Event(finished) attribute on it that shows that event "finished" should be handled by finish method. If event name is the same as method name you can use just :Event attribute without parameters, as for "arrival" event in this example.

Here are two types of events: "arrival" - that means that new customer has arrived, and "finished" that means that server has finished serving customer. Let's start from 'arrival' handler. First we're computing time when next customer will arrive, for that we're adding to current model time ($self->model->time) random value with exponential distribution. After that we're scheduling event for the next customer. Schedule function accepts three arguments (fourth argument is optional): time of the event to schedule, reference to the server that should handle event, and type of the event. After scheduling event for the next customer we're returning to current and checking if server busy or not. If server already busy, then we reject arrived customer. If server is free, customer is served. We're computing time when serving will be finished and scheduling that event. Also, we're changing server state to busy.

Handler for "finished" event is much simpler. We're just increasing number of served customers and changing server state back to free.

Completing Model

Now we have a class for server and can return to the model. In order to add server to model we're using the add method:

    $server = $model->add(
        "Simulation::DiscreteEvent::CB::MM10",
        lambda => 2,
        mu     => 3,
    );

This will create new instance of Simulation::DiscreteEvent::CB::MM10 with lambda and mu passed to constructor.

We almost ready to run. Just one detail: before we start simulation we should schedule at least one event. Without it simulation will finish immediately. We could use schedule method, which we used in server code, but instead let's try another method - send. This method schedules event for current time. As we're not yet started simulation the current model time is 0.

    $model->send($server, "arrival");

So after we start simulation, server will receive "arrival" event, will start serving customer, and schedule next "arrival" and "finished" events. In order to start simulation we're using run method.

    $model->run(1000);

Simulation runs while there are events scheduled, and while model time is less than value passed to run method. In our case simulation will be stopped at model time 1000.

And after simulation finished we print numbers of served and rejected customers, and loss probability:

    print "Served customers:    ", $server->served, "\n";
    print "Rejected customers:  ", $server->rejected, "\n";
    print "Customers loss rate: ", 
        $server->rejected/($server->served + $server->rejected), "\n";

And here's the whole script:

    use strict;
    use warnings;
    use Simulation::DiscreteEvent;
    
    my $model = Simulation::DiscreteEvent->new;
    
    my $server = $model->add(
        "Simulation::DiscreteEvent::CB::MM10",
        lambda => 2,
        mu     => 3,
    );
    
    $model->send( $server, "arrival" );
    $model->run(1000);
    
    print "Served customers:    ", $server->served,   "\n";
    print "Rejected customers:  ", $server->rejected, "\n";
    print "Customers loss rate: ",
        $server->rejected / ( $server->served + $server->rejected ), "\n";

This will output something like that:

    Served customers:    1221
    Rejected customers:  820
    Customers loss rate: 0.401763841254287

So we can see that about 40% of customers will be lost, and in real life it probably means that we should increase throughput of the server, or add more servers.

AUTHOR ^

Pavel Shaydo, <zwon at cpan.org>

BUGS ^

Please report any bugs or feature requests to bug-simulation-discreteevent at rt.cpan.org, or through the web interface at http://rt.cpan.org/NoAuth/ReportBug.html?Queue=Simulation-DiscreteEvent. I will be notified, and then you'll automatically be notified of progress on your bug as I make changes.

SEE ALSO ^

Simulation::DiscreteEvent

LICENSE AND COPYRIGHT ^

Copyright 2010 Pavel Shaydo.

This program is free software; you can redistribute it and/or modify it under the terms of either: the GNU General Public License as published by the Free Software Foundation; or the Artistic License.

See http://dev.perl.org/licenses/ for more information.

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