1. Queueing Theory What is a queue? Examples of queues:
Grocery store checkout
Fast food (McDonalds – vs- Wendy’s)
Hospital Emergency rooms
Machines waiting for repair
Communications network

2. Queueing Theory Basic components of a queue:
customers
Input source Queue Service
(calling population) mechanism

3. Queueing Theory Input source: population size (assumed infinite)
customer generation pattern (assumed Poisson w rate or equivalently, exponential with an inter-arrival time )
arrival behavior (balking, blocking)
Queue:
queue size (finite or infinite)
queue discipline (assumed FIFO, other include random, LIFO, priority, etc..)

4. Queueing Theory Service mechanism: number of servers
service time and distribution (exponential is most common)

5. Queueing Theory Naming convention: a / b / c
a – distribution of inter-arrival times
b – distribution of service time
c – number of servers
Where M – exponential distribution (Markovian)
D – degenerate distribution (constant times)
Ek – Erlang distribution (with shape k)
G = general distribuiton
Ex. M/M/1 or M/G/1

6. Queueing Theory Terminology and Notation:
State of system – number of customers in the queueing system, N(t)
Queue length – number of customer waiting in the queue
- state of system minus number being served
Pn(t) – probability exactly n customers in system at time t.
s – number of servers
ln – mean arrival rate (expected arrivals per unit time) when n customers already in system
mn – mean service rate for overall system (expected number of customers completing service per unit time) when n customers are in the system.
r – utilization factor (= l/sm , in general)

7. Queueing Theory Terminology and Notation:
L = expected number of customers in queueing system =
Lq = expected queue length =
W = expected waiting time in system (includes service time)
Wq = expected waiting time in queue

8. Queueing Theory Little’s Law: W = L / l and Wq = Lq / l
Note: if ln are not equal, then l = l
W = Wq + 1/m when m is constant.

9. Role of Exponential Distribution
Let T be a random variable
representing the inter-arrival
time between events.
fT(t) a
Property 1: fT(t) is a strictly
decreasing function of t (t > 0).
P[0 < T < ] > P[t < T < t ]
P[0 < T < 1/2a] = .393
P[1/2a < T < 3/2a] = .383
Property 2: lack of memory.
P[T > t | T > ] = P[T > t] t
fT(t) = ae-at for t > = 0
FT = 1 - e-at
E(T) = 1/a
V (T) = 1/a2

10. Role of Exponential Distribution
Property 3: the minimum of several independent exponential random variables has an exponential distribution.
Let T1, T2,…. Tn be distributed exponential with parameters a1, a2,…, an and let U = min{T1, T2,…. Tn } then:
U is exponential with rate parameter a =
Property 4: Relationship to Poisson distribution.
Let the time between events be distributed exponentially with
Parameter a. Then the number of times, X(t), this event occurs over some time t has a Poisson distribution with rate at:
P[X(t) = n] = (at)ne-at/n!

11. Role of Exponential Distribution
Property 5: For all positive values of t, P[T < t | T > t]
for small a can be thought of as the mean rate of occurrence.
Property 6: Unaffected by aggregation or dis-aggregation.
Suppose a system has multiple input streams (arrivals of customers) with rate l1, l2,…ln, then the system as a whole has
An input stream with a rate l =

12. Break for Exercise

13. Birth-Death Process Birth – arrival of a customer to the system
Death – departure of a customer from the system
N(t) – random variable associated with the state of the system at time t (i.e. the number of customers, n, in the system at time t).
Assumptions – customers inter-arrival times are exponential at a rate of ln and their service times are exponential at a rate of mn
Queuing System
Departures - mn
Arrivals - ln

14. Rate Diagram for Birth-Death Process1 2
n-2
n-1 n … …

15. Development of Balance Equations for Birth-Death Process
M/M/1 System 1 2
n-2
n-1 n … …

16. Development of Balance Equations for Birth-Death Process
M/M/s System – multiple servers 1 2
n-2
n-1 n … …

17. Development of Balance Equations for Birth-Death Process
M/M/1/k System – finite system capacity 1 2
k-2
k-1 k …

18. Development of Balance Equations for Birth-Death Process
M/M/1 System – finite calling system population 1 2
N-2
N-1 N …

19. M/G/1 Queue
A system with a Poisson input, but some general distribution
for service time.
Only two characteristics of the service time need be known:
the mean (1/m) and the variance (1/s2).
P0 = 1 – r, where r = l/m
Lq =
L = r + Lq
Wq = Lq / l W = Wq + 1 / m