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Question 11 – 3. Question 11 – 9. Question 11 – 9 cont. Question 11 – 11 a. Question 11 – 11 b, c. Queueing Theory: Part II. Elementary Queueing Process. Served customers. Queueing system. Queue. C C C C. S S Service S facility S. Customers. C C C C C C C.

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slide7

Elementary Queueing Process

Served customers

Queueing system

Queue

C

C

C

C

S

S Service

S facility

S

Customers

C C C C C C C

Served customers

slide8

Relationships between and

Assume that is a constant for all n.

In a steady-state queueing process,

Assume that the mean service time is a constant,

for all It follows that,

slide9

The Birth-and-Death Process

Most elementary queueing models assume that the inputs and outputs of the queueing system occur according to the birth-and-death process.

In the context of queueing theory, the term birth refers to the arrival of a new customer into the queueing system, and death refers to the departure of a served customer.

slide10

The birth-and-death process is a special type of continuous time Markov chain.

State: 0 1 2 3 n-2 n-1 n n+1

and are mean rates.

slide11

Rate In = Rate Out Principle.

For any state of the system n (n = 0,1,2,…),

average entering rate = average leaving rate.

The equation expressing this principle is called the balance equation for state n.

slide12

Rate In = Rate Out

State

0

1

2

n – 1

n

slide13

State:

0:

1:

2:

To simplify notation, let

for n = 1,2,…

slide14

and then define for n = 0.

Thus, the steady-state probabilities are

for n = 0,1,2,…

The requirement that

so that

implies that

slide15

The definitions of L and specify that

is the average arrival rate. is the mean arrival rate while the system is in state n. is the proportion of time for state n,

slide16

The Finite Queue Variation of the M/M/s Model]

(Called the M/M/s/K Model)

Queueing systems sometimes have a finite queue; i.e., the number of customers in the system is not permitted to exceed some specified number. Any customer that arrives while the queue is “full” is refused entry into the system and so leaves forever.

slide17

From the viewpoint of the birth-and-death process, the mean input rate into the system becomes zero at these times.

The one modification is needed

for n = 0, 1, 2,…, K-1

for n K.

Because for some values of n, a queueing system that fits this model always will eventually reach a steady-state condition, even when

slide18

Question 1

Consider a birth-and-death process with just three attainable states (0,1, and 2), for which the steady-state probabilities are P0, P1, and P2, respectively. The birth-and-death rates are summarized in the following table:

State

Birth Rate

Death Rate

0

1

2

1

1

0

_

2

2

slide19

Construct the rate diagram for this birth-and-death process.

  • Develop the balance equations.
  • Solve these equations to find P0,P1, and P2.
  • Use the general formulas for the birth-and-death process to calculate P0 ,P1 , and P2. Also calculate L, Lq, W, and Wq.
slide20

Question 1 - SOLUTINON

Single Serve & Finite Queue

(a) Birth-and-death process

0

1

2

(b)

In Out

(1)

Balance

Equation

(2)

(3)

(4)

slide24

Question 2

  • Consider the birth-and-death process with the following mean rates. The birth rates are =2, =3, =2, =1, and =0 for n>3. The death rates are =3 =4 =1
  • =2 for n>4.
  • Construct the rate diagram for this birth-and-death process.
  • Develop the balance equations.
  • Solve these equations to find the steady-state probability distribution P0 ,P1, …..
  • Use the general formulas for the birth-and-death process to calculate P0 ,P1, ….. Also calculate L ,Lq, W, and Wq.
slide25

Question 2 - SOLUTION

(a)

3

4

0

1

2

(b)

(1)

(2)

(3)

(4)

(5)

(6)