1. Mitchell Jareo MAT4340 – Operations Research Dr. Bauldry
Queueing Theory
Mitchell Jareo
MAT4340 – Operations Research
Dr. Bauldry

2. Queue Characteristics
5 Characteristic Components
Arrival Pattern
Service Pattern
Number of Servers
Service Facility Capacity
Service Order

3. Arrival Patterns Interarrival Time Balking – queue is too long
Time between customer arrivals to the facility
Deterministic
Random by a known probability distribution
May be state dependent or not
Single arrivals or in batches
Balking – queue is too long
Reneging – service wait is too long

4. Service Patterns Service Time
Time for one server to serve one customer
Deterministic
Random by a known probability distribution
May be state dependent or not
One server or sequence of servers

5. System Capacity
Maximum number of customers service and in queues which the facility can hold
Customers cannot enter full facilities
Cannot wait outside; must leave
Capacity can be finite or infinite

6. Queue Disciplines Order in which customers are served
FIFO (first-in, first-out) order of arrival
LIFO (last-in, last-out) last one first
Random basis
Priority basis
Whatever other basis you can cook up

7. Kendall’s Notation Specifies queue characteristics v/w/x/y/z where
v = interarrival pattern (D/M/Ek/G)
w = service pattern (D/M/Ek/G)
x = number of available servers
y = system capacity
z = queue discipline (FIFO,LIFO,SIRO,PRI,GD)
y and z default to infinite and FIFO

8. Examples Grocery Store with 6 checkout lanes
M/M/6/24/FIFO
Car Wash with 4 car waiting lane
M/M/1/4/FIFO
Work for a 12 typist typing pool
M/M/12/500/LIFO
Emergency Room w/2 docs and 20 seats
M/M/2/20/PRI

9. A Deterministic Example
A bus cleaning facility. D/D/1
Buses arrives one the hour five at a time
Cleaning takes 11 minutes
Simulated queue for one hour
Average buses in facility
Average queue length
Average time in facility

10. M/M/1 Systems
Exponentially distributed interarrival times with parameter, λ.
Exponentially distributed service times with parameter, μ.
1 server.
λ is the average customer arrival time
μ is the average service rate.
Both are customers per unit time.

11. M/M/1 Systems
Given the exponentially distributed interarrival and service times.
Expected interarrival time = 1/ λ
Expected serice time = 1/ μ
Utilization factor, or traffic intensity
Expected number of arrivals per mean service time is denoted as ρ = λ/ μ
If ρ < 1, then steady state probabilities exist and are given by ρ n = ρ n (1 – ρ)

12. M/M/1 Measures L – average number of customers
Lq – average length of queue
W – average time customer is in system
Wq – average customer queue wait time
W(t) – probability customer spends more than t time units in system
Wq (t) – probability customer spends more than t time units in queue