Now I am moving into M/M/c/K model. So here the change is instead of one server in the
M/M/1 model you have more than one servers c and you have a finite capacity that is K,
capacity of the system. So the arrival follows the Poisson process, service is exponential,
we have c identical servers. The capacity is K and this is the scenario in which whenever
the system size is less than c it will be routed into the ideal server.
And if it is greater than or equal to c, that means all the servers are busy that means
the customer has to wait. But if the system size is full, that means c customers are under
service and K minus c customers are waiting in the queue for the service and then whoever
comes it will be rejected force to leave the system.
And therefore you have a waiting as well as blocking because it is a finite capacity there
is blocking and since you have, always we choose K such that it is K is always greater
than or equal to c. If K is equal to c, then it is a loss system and then if K is greater
than c, then K minus c customers maximum can wait in the system, in the queue.
Therefore, the underlying stochastic process, here the stochastic process is again number
of customers in the system at any time t. Therefore, this stochastic process is also
going to be a continuous time Markov chain because of these assumptions, inter arrivals
are exponential distribution service, each service by each server is exponentially distributed
and all are independent and so on.
So with these assumptions these stochastic processes is a continuous time Markov chain
and at any time only, only one forward or only one backward the system can move. Therefore,
it is going to be a birth death process also and the birth rates are lambda because it
is a infinite source population, so all the lambda n are going to be lambda, whereas the
death rates are state dependent.
That is going to be n times mu, lies between 1 to c, from c to k onwards it is going to
be c mu. I have not drawn the state transition diagram for M/M/c/K, but you can visualize
the way we have M/M/1/N and M/M/c model, so it is a combination of that, that is going
to be the state transition diagram. Since it is a finite capacity model, it is easy
to get the steady state and the equilibrium solution.
So first you solve pi Q is equal to zero that means you write pi n in terms of pi 0 and
use a normalizing constant summation of pi i is equal to one, using that you will get
pi 0. So I have not written here. So use the normalizing constant summation of pi i is
equal to one, get the pi 0, then substitute pi 0 here therefore you will get pi n in terms
of pi 0 completely.
After that you can get all other average measures, the way I have explained M/M/1/N and the M/M/c
infinity, the combination of that you can get the average number of customers in the
system, average number of customers in the queue. That is n minus c times pi n, combination,
the summation goes from c to K and the average time spend in this system.
Since it is a finite capacity you have to find out the lambda effective, effective arrival
rate, that his one minus, its capacity is capital K, therefore one minus pi K and that
is the probably that the system is not full. So the effective arrival rate is lambda times
one minus pi K, substitute here and get the average time spent in the system and similarly
you can find out the average time spent in the queue also using the little’s formula.
Now I am moving into the fourth simple Markovian queuing model. First I started with the M/M/c
infinity, M/M/1/N, then I did M/M/c/K and now I am going for K is equal to c that is
loss system. It is not a queuing system because we have c servers and the capacity of the
system is also c.
Example is you can think of parking lot which has some c parking lots and the cars coming
into the system that is if you make the assumption is inter arrival time is exponentially distributed
and the car spending time in each parking lot that is exponentially distributed then
the parking lot problem can be visualized as the M/M/c loses too. So here we have c
identical servers, no waiting room.
So since it is c capacity and c waiting room you think of self service with the capacity
c, that also you can visualize. So here the inter arrival times are exponentially distributed
and service by each server, that is exponentially distributed with the parameter mu therefore
the system goes from two to one, one to zero, so on, it is going to be how many customers
in the system and completing their service.
Therefore, the time is exponentially distributed with the sum of those parameters accordingly.
Therefore, it is going to be 1mu, 2 mu till c mu. Since it is a finite capacity and so
on, it is a reducible model, positive recurrent. So this listed probability exists, limiting
probability also exists and that is same as the equilibrium probabilities also. Therefore,
by using pQ is equal to zero and summation of p i is equal to one you can get the steady
state or the equilibrium probabilities, that is p n.
The p suffix c, that is nothing but the probability that the system is full and that is same known
as the Erlang B formula. So this is also useful to design the system for a given or what is
the optimal c such that you can minimize the probability that the system is full, for that
you need this formula therefore to do the optimization problem over the c and here we
denote p suffix c, that is Erlang B formula. Whereas Erlang C formula comes from the M/M/c/K
model for the last system we will get the Erlang B formula.
The fifth model, that is M/M/∞, it is not a queuing model, because servers are infinite
and limited servers in the system and therefore the customers whoever enters he will get it
immediately serviced. The service will be started immediately whereas the service time
is exponentially distributed with the parameter mu by each server, all the servers are identical.
The number of servers are infinite here. Therefore, you will have the underlying stochastic process
for the system size that is a birth death process with the birth rates are lambda because
the population is from the infinite source, the death rates are 1 mu, 2 mu and so on because
the number of servers are infinite. So the model which I have discussed in today’s
lecture, all the five models are the underlying stochastic processes, birth death process.
This is simplest Markovian queuing models.
You can get the steady state distribution, use the same theory of birth death process
and if you observe these steady state probabilities is of the same Poisson, it is of the form
that is probability mass function of a Poisson distribution. Therefore, you can conclude
in a steady state, number of customers in the system that is Poisson distributed with
the parameter lambda by mu.
Because the probability mass function for the pi n is same as the probability mass function
of exponentially distributed random variable with the parameter lambda by mu.