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

2. 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
The service facility may contain more than one server, and it is assumed that a server can serve one customer at a time. If an arriving customer finds the service facility occupied, he joins the waiting queue. This customer will receive his service later in time, either when he reaches the head of the waiting queue or according to some service disciplines, and then leave the system upon completion of his service. The schematic diagram of a queuing system is depicted in the above presented figure. A queuing system is referred to as just a queue, or queuing node.
Customer
Population
Waiting
queue
Service
facility

3. Characteristics of the system structure
λ: arrival rate
μ: service rate λ μ
Queue
Infinite or finite
Service mechanism
1 server or S servers
Queuing discipline
FIFO, LIFO, priority-aware, or random
The service facility may comprise one or more servers. Queuing discipline refers to the way in which customers in the waiting queue are selected and served by the servers. In general we have: First Come First Served (FCFS), Last Come First Served (LCFS), priority, or random.

4. Queuing systems: examples
Multi queue/multi servers
Example:
Supermarket
Blade centers
orchestrator
Multi-server/single queue
Bank
immigration .

5. 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

6. 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

7. Special queuing systems
Infinite server queue
Machine interference (finite population) λ μ .
S repairmen N
machines

8. M/M/1 queue λn = λ, (n >=0); μn = μ (n>=1) λ μ λ: arrival rate
μ: service rate
λn = λ, (n >=0); μn = μ (n>=1)

9. 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
To understand better the physical meaning of this unit, take a look at the traffic presented to a single resource. One Erlang is equivalent to a single user who uses that resources 100% of the time, or alternatively, 10 users who each occupy the resource 10% of the time. A traffic intensity greater than one indicates that customers arrive faster than they are served and is a good indication of the minimum number of servers required to achieve a stable system. For an example, a traffic intensity of 2.5 Erlangs indicates that at least three servers are required.

10. Queuing systems: stability
N(t)
λ<μ
=> stable system
λ>μ
Steady build up of customers => unstable
busy
idle 1 2 3
Time
N(t) 1 2 3
Time

11. 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

12. Performance measures L Lq W Wq Mean # customers in the whole system
Mean queue length in the queue space W
Mean waiting time in the system Wq
Mean waiting time in the queue

13. Mean queue length (M/M/1)

14. Mean queue length (M/M/1) (cont’d)