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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
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

system

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,

m m m queue
M/M/m Queue

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

m m m m queue
M/M/m/m/ Queue

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
Multi-Dimensional Markov Chain

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

transition probability diagram
Transition Probability Diagram

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

k,0

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
Truncation of Multi Dimensional System

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

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
Important Results for M/G/1 Queue

Pollaczek=Khinchin Formula: