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

Queueing Theory

Basic components of a queue:

customers

Input source Queue Service(calling population) mechanism

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..)

Queueing Theory

Service mechanism:

• number of servers

• service time and distribution (exponential is most common)

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

Queueing TheoryTerminology 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

n – mean arrival rate (expected arrivals per unit time) when n customers already in system

n – mean service rate for overall system (expected number of customers completing service per unit time) when n customers are in the system.

– utilization factor (= s, in general)

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

0nnnP

sn

nPsn )(

Queueing Theory

Little’s Law:

W = L /

and

Wq = Lq /

Note: if n are not equal, then =

W = Wq + 1/when is constant.

0nnnP

Role of Exponential Distribution

Property 1: fT(t) is a strictly

decreasing function of t (t > 0).

P[0 < T < ] > P[t < T < t + ]

P[0 < T < 1/2] = .393

P[1/2 < T < 3/2] = .383

Property 2: lack of memory.

P[T > t + | T > ] = P[T > t]

fT(t) = e-t for t > = 0

FT = 1 - e-t

E(T) = 1/

V (T) = 1/2

fT(t)

t

t t

tt

Let T be a random variable representing the inter-arrivaltime between events.

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 1, 2,…, n and let U = min{T1, T2,…. Tn } then:

U is exponential with rate parameter =

Property 4: Relationship to Poisson distribution.

Let the time between events be distributed exponentially with

Parameter . Then the number of times, X(t), this event occurs over some time t has a Poisson distribution with rate t:

P[X(t) = n] = (t)ne-t/n!

n

ii

1

Role of Exponential Distribution

Property 5: For all positive values of t, P[T < t + | T > t]

for small . 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 1, 2,…n, then the system as a whole has

An input stream with a rate =

tt

t

n

ii

1

Break for Exercise

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 n and their service times are exponential at a rate of n

Arrivals - nDepartures - n

Queuing System

Rate Diagram for Birth-Death Process

0 1 2 n-1n-2 n… …

Development of Balance Equations for Birth-Death Process

M/M/1 System

0 1 2 n-1n-2 n… …

Development of Balance Equations for Birth-Death Process

M/M/s System – multiple servers

0 1 2 n-1n-2 n… …

Development of Balance Equations for Birth-Death Process

M/M/1/k System – finite system capacity

0 1 2 k-1k-2 k…

Development of Balance Equations for Birth-Death Process

M/M/1 System – finite calling system population

0 1 2 N-1N-2 N…

M/G/1 Queue

A system with a Poisson input, but some general distributionfor service time.

Only two characteristics of the service time need be known:the mean (1/) and the variance (1/2).

P0 = 1 – , where = /

Lq =

L = + Lq

Wq = Lq / W = Wq + 1 /

)1(2

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