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Infinite Capacity Markovian Models
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M/M/1/

The M/M/1 is a single-server queueing model, which can be used to approximate simple systems.  The M/M/1 queuing system is described as a queuing model where:

 

§  Arrivals form a Poisson process i.e. interarrival time is exponentially distributed

§  Service time is exponentially distributed

§  There is one server

§  The length of queue in which arriving users wait before being served is infinite

§  The population of users (i.e. the pool of users) available to join the system is infinite

A simple M/M/1 queue with arrival rate λ and service rate µ

There are a lot of situations where an M/M/1 model could be used. For instance, you can take a post office with only one employee, and therefore one queue. The customers arrive, go into the queue, they are served, and they leave the system. If the arrival process is Poisson, and the service time is exponential, we can use an M/M/1 model. Hence, we can easily calculate the expected number of people in the queue, the probabilities they will have to wait for a particular length of time, and so on.

Steady State Distribution

A birth and death process is a M/M/1 queue when λi = λ and μi = μ for all i.

Let pn represents the probability mass function of a discrete random variable denoting the number of customers in the system in long run  

                                                       pn = (1-ρ) ρn, ρ<1

where,

 ρ = λ/µ represents the traffic intensity of the system. For a stable system the intensity ρ must be less than 1.

It can be seen above that the steady state probabilities for an M/M/1 queue follows the geometric distribution with parameter (1- ρ).

Measures of Effectiveness

Measure

Expression

Average number of customers in the system (Ls)

ρ/(1- ρ)

Average number of customers in the Queue (Lq)

ρ2/(1- ρ)

Expected waiting time in system (W)

1/(µ-λ)

Expected waiting time in queue (Wq)

ρ/( µ-λ)

Utilization

Ρ

 

Transient solution

The transient probabilities pn(t) = Pr {X(t) = n} for an M/M/1 queue are given by

for all n ≥ 0, where

is the infinite series for the modified Bessel function of the first kind.

M/M/c/

M/M/c model is a multi-server queue model. It is a generalization of the M/M/1 model.

The M/M/c queuing system is described as a queuing model where:

  • Arrivals are a Poisson process
  • Service time is exponentially distributed
  • There are c servers
  • The length of queue in which arriving users wait before being served is infinite
  • The population of users (i.e. the pool of users), or requests, available to join the system is infinite

Steady State Distribution

M/M/c queue differs from the M/M/1 queue only in the service time which now becomes,

μi  = iμ for i <= c

μi = cμ for i >= c

λi = λ for all i.

 

For this model the steady state probabilities are given by:

where,

 ,        ρ<1

                                                          

p = λ/cµ,  r = ρ = λ/µ

 

Measures of Effectiveness

Measure

Expression

Average number of customers in the system (Ls)

Average number of customers in the Queue (Lq)

Expected waiting time in system (W)

Expected waiting time in queue (Wq)

 

 

 

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