#### Transcript ppt - Dr. Wissam Fawaz

**Introduction**

Definition M/M queues M/M/1 M/M/S M/M/infinity M/M/S/K 1

**Queuing system**

A queuing system is a place where customers arrive According to an “arrival process” To receive service from a service facility Can be broken down into three major components The input process The system structure The output process Customer Population Waiting queue Service facility 2

λ

**Characteristics of the system structure**

μ λ: arrival rate μ: service rate Queue Infinite or finite Service mechanism 1 server or S servers Queuing discipline FIFO, LIFO, priority-aware, or random 3

**Queuing systems: examples**

Multi queue/multi servers Example: Supermarket Blade centers orchestrator .

.

.

Multi-server/single queue Bank immigration 4

**Kendall notation**

David Kendall A British statistician, developed a shorthand notation To describe a queuing system A/B/X/Y/Z A: Customer arriving pattern B: Service pattern X: Number of parallel servers Y: System capacity Z: Queuing discipline

**M: Markovian D: constant G: general Cx: coxian**

5

**Kendall notation: example**

M/M/1/infinity A queuing system having one server where Customers arrive according to a Poisson process Exponentially distributed service times M/M/S/K K M/M/S/K=0 Erlang loss queue 6

**Special queuing systems**

Infinite server queue λ .

.

μ Machine interference (finite population)

**N machines**

S repairmen 7

λ

**M/M/1 queue**

μ λ: arrival rate μ: service rate λ n = λ, (n >=0); μ n = μ (n>=1)

*P n*

0 0 1 1 ...

...

*n*

1

*n P*

0

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0 ;

*P*

0

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0 ( 1 2 ...

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*P*

0 1 8

**Traffic intensity**

rho = λ/μ It is a measure of the total arrival traffic to the system Also known as offered load Example: λ = 3/hour; 1/μ=15 min = 0.25 h Represents the fraction of time a server is busy In which case it is called the utilization factor Example: rho = 0.75 = % busy 9

**Queuing systems: stability N(t)**

λ<μ => stable system

**busy idle 3 2 1**

λ>μ

**1 2 3 4 5 6 7 8 9 10 11 Time **

Steady build up of customers => unstable

**N(t) 3 2 1 1 2 3 4 5 6 7 8 9 10 11 Time **

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**Example#1**

A communication channel operating at 9600 bps Receives two type of packet streams from a gateway Type A packets have a fixed length format of 48 bits Type B packets have an exponentially distribution length With a mean of 480 bits If on the average there are 20% type A packets and 80% type B packets Calculate the utilization of this channel Assuming the combined arrival rate is 15 packets/s 11

**Performance measures**

L Mean # customers in the whole system L q Mean queue length in the queue space W Mean waiting time in the system W q Mean waiting time in the queue 12

**Mean queue length (M/M/1)**

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**Mean queue length (M/M/1) (cont’d)**

*L q*

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14