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Finite M/M/1 queue Consider an M/M/1 queue with finite waiting room. (The previous result had infinite waiting room) We can have up to packets in the system. After filling the system, packets are returned, or blocked. Balance equations: 0 1

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Finite m m 1 queue l.jpg
Finite M/M/1 queue

Consider an M/M/1 queue with finite waiting room.

(The previous result had infinite waiting room)

We can have up to packets in the system.

After filling the system, packets are returned, or blocked.

Balance equations:

0

1



Slide3 l.jpg

system particular interest is

Consider any queue with blocking probability PB and load  packets/second.

Net arrival rate = (1- PB) . Then  = (1- PB)  = throughput.

From a different point of view,


Slide4 l.jpg

For M/M/1 queue of finite length, particular interest is


M m m queue l.jpg
M/M/m Queue particular interest is

There are m servers and the customers line up in one queue. The customer at the head of the queue is routed to the available server.

Balance equations:

0

1

……….

m+1

m+2

m-1

m


Erlang c formula l.jpg
Erlang C Formula particular interest is


M m queue l.jpg
M/M/ particular interest is Queue


M m m m queue l.jpg
M/M/m/m/ Queue particular interest is

There are m servers. If a customer upon arrival finds all servers busy, it does not enter the system and is lost. The m in •/•/•/m is the limit of the number of customers in the system. This model is used frequently in the traditional telephony. To use in the data networks, we can assume that m is the number of virtual circuit connections allowed.

Balance equations:

….

m-1

m

0

1


Multi dimensional markov chain l.jpg
Multi-Dimensional Markov Chain particular interest is

Consider transmission lines with m independent circuits of equal capacity. There are two types of sessions:


Transition probability diagram l.jpg
Transition Probability Diagram particular interest is

m, 0

m-1, 0

m-1, 1

.

.

.

.

.

.

.

.

.

1, m-1

1,0

1,2

1,1

.

.

.

0, m-1

0, m

0,2

0,1

0,0


Slide14 l.jpg

k,0 particular interest is

k,1

k, m-k

….

Suppose in the previous case, there is a limit k < m on the number of circuits that can be used by sessions of type 2.

k-1, 2

k, m-k

k-1,0

k-1, 1

….

k-1, m-k-1

….

….

….

….

….

1, m-1

1,0

….

1,2

1,1

0, m

0, m-1

….

0,2

0,1

0,0

Blocking probabilities for call types:



Truncation of multi dimensional system l.jpg
Truncation of Multi Dimensional System dimensional Markov Chain.

Consider l M/M/1 in independent queues.

Then, for the joint queue, the following is true:

Above is also true for M/M/m, M/M/, M/M/m/m, and all other birth-death processes.

We now consider truncation of multi dimensional Markov Chain. Truncation is achieved by eliminating (or not considering) some of the states with low probability. The truncated system is a Markov Chain with the same transition diagram without some of the states that have been eliminated.


Slide17 l.jpg

Claim: Stationary distribution of the truncated system is in a product form.

Proof: We have detailed balance equations:

Substituting we can show that balance

equations hold true with

Since the solution satisfies the balance equations,

it must the unique stationary distribution.



Important results for m g 1 queue l.jpg
Important Results for M/G/1 Queue a product form.

Pollaczek=Khinchin Formula:


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