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Introduction

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

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

Service

facility

Waiting

queue

Characteristics of the system structure

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

λ

μ

Queuing systems: examples

- Multi queue/multi servers
- Example:
- Supermarket
- Blade centers
- orchestrator
- Multi-server/single queue
- Bank
- immigration

.

.

.

D: constant

G: general

Cx: coxian

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

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
- M/M/S/K=0
- Erlang loss queue

K

Special queuing systems

- Infinite server queue
- Machine interference (finite population)

μ

λ

.

.

S repairmen

N

machines

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

3

2

2

1

1

busy

idle

Queuing systems: stabilityN(t)

- λ<μ
- => stable system
- λ>μ
- Steady build up of customers => unstable

1 2 3 4 5 6 7 8 9 10 11

Time

N(t)

1 2 3 4 5 6 7 8 9 10 11

Time

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

Performance measures

- L
- Mean # customers in the whole system
- Lq
- Mean queue length in the queue space
- W
- Mean waiting time in the system
- Wq
- Mean waiting time in the queue

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