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Finite Capacity Markovian Models
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Finite Capacity Markovian Queuing Models

M/M/1/N

The M/M/1/N queue is a single server queue with a buffer of size N. This queue has applications in telecommunications, as well as in biology when a population has a capacity limit.

Steady State Distribution

We again use the parameters from the M/M/1 queue with,

λi = λ  for  0 <= i< N

λi = 0  for  i >= N

μi = μ  for 1 <= i <= N.

 

The state probabilities in equilibrium are given by:

                                             

Measures of Effectiveness

Measure

Expression

Average number of customers in the system (Ls)

Average number of customers in queue (Lq)

Ls – (λ/µ)

 

Expected waiting time in system (W)

Ls/ λ

Expected waiting time in queue (Wq)

Lq/ λ

Utilization

ρ

Blocking Probability (PB)

Throughput

ρ (1 -  PB )

 

 

 

 

 

 M/M/c/N

The M/M/c/N queue is a multi server queue with a buffer of size N.

Steady state distribution

μi  = iμ  for  i <= c

μi = cμ  for  c <= i <= N

λi = λ for all i.

 

The steady state system size probabilities are given by:

For this model the steady state probabilities are given by:

                                             

where,

                                            

where,  ,

 

Measures of Effectiveness

 

Measure

Expression

Average number of customers  in the Queue (Lq)  

 

,  ρ≠1

Average number of customers  in the system (Ls)  

Lq + r(1-PN)

Expected waiting time in system (W)

Expected waiting time in queue (Wq)

Utilization

ρ

Blocking Probability (PB)

Throughput

 

 

M/M/c/c

This is a special case of the truncated queue M/M/c/N for which N = c, i.e. where no queue is allowed to form.  This is also known as Erlang loss system. It plays an important role in telecommunication.

Steady State Distribution

For this model the steady state probabilities are given by:

                                                        

In case of an M/M/c model we define the following performance measures:

Measure

Expression

Blocking Probability (PB)

Throughput

ρ (1 -  PB )

 

 where , r = (λ/µ)

 

 

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