Networking Theory and Fundamentals - Lecture 8 potx

8-2 Topics„ Closed Jackson Networks „ Convolution Algorithm „ Calculating the Throughput in a Closed Network „ Arrival Theorem for Closed Networks „ Norton’s Equivalent... 8-5 Closed Jac

TCOM 501:

Networking Theory & Fundamentals

Lecture 8 March 19, 2003 Prof Yannis A Korilis

8-2 Topics

„ Closed Jackson Networks

„ Convolution Algorithm

„ Calculating the Throughput in a Closed Network

„ Arrival Theorem for Closed Networks

„ Norton’s Equivalent

8-3 Closed Jackson Networks

1

µ

2 λ

λ

5

λ 51

r

53

r

M

„ Closed Network: K nodes with exponential servers

„ No external arrivals (γi =0) , no departures (r i0=0)

„ Fixed number M of circulating customers

„ Appropriate model for systems with “limited” resources, e.g., flow control mechanisms

Steady-state distribution will be shown to be of “product-form” type

8-4 Closed Jackson Network

„ Aggregate arrival rates

Relative arrival rates – visit ratios

„ Can only be determined up to a constant

„ Use an additional equation to obtain unique solution to the above system, e.g.

„ Set λj =1, for some node j

„ Set λj=µj , for some node j

„ Set λ1+ λ2+…+ λK=1

„ n i : number of customers at node i

„ Possible states for the closed network n=(n1, n2,…,n K), with

„ Let F(M) denote the set of all such states

8-5 Closed Jackson Network

„ Let x i be the number of customers at station i, at steady state

„ Random variables x1, x2,…, x K are not independent – their sum must be

equal to M

„ The state x=(x1, x2,…, x K) of the closed network can take values

n=(n1, n2,…,n K), with

„ Let F(M) denote the set of all such states

„ Define ρi ≡ λi/µi – this is not the actual utilization factor of station i

„ Jackson’s theorem for closed networks gives the stationary distribution

8-6 Jackson’s Theorem for Closed Networks

„ Theorem 1: The stationary distribution of a closed Jackson network is

where the normalization constant G(M) is a function of M

„ G(M) guarantees that {p(n)} is a valid probability distribution

„ This stationary distribution is said to have a product-form

„ However: at steady-state the queues are not independent

„ {p i (n i)}: marginal stationary distribution of queue i

8-7 Jackson’s Theorem for Closed Networks (proof)

„ Theorem 2: The reversed chain of a stationary closed Jackson network

is also a stationary closed Jackson network with the same service rates and routing probabilities:

„ Proof of Theorems 1 & 2:

Show that for the corresponding forward and reversed chains

„ Need to prove only for m=T ij n

Verify, exactly as in the open-network case, that:

8-8 Closed Jackson Network

Example: Closed network model for CPU (rate µ1) and

I/O (rate µ2) system Upon service completion in 1,

customer routed to 2 with probability p2=1-p1, or back

to 1 with probability p1 M =fixed number of customers

1 p1 1 2 , 2 p2 1 Choose solution: 1 1 and 2 p2 1

0

2

1 ( )

1

M

M n n

2

p

8-9 Closed Networks: Normalization Constant

„ Normalization constant as a function of M and K:

All performance measures of interest – throughput, average delay – can

be obtained in terms of G(M,K)

„ Computational complexity is exponential in M and K:

„ Recursive methods can be used to reduce complexity

„ Iterative algorithm [due to Buzen]

Normalization constant will be treated as a function of both M and K

and denoted G(M,K) only in the context of the iterative algorithm

1 2 1

1 2 ( ) 1

0

K i

M K M

+ −

8-10 Iterative Computation of G(M)

„ For any m and k (m=0,…, M; k=1,…, K) define:

„ For a closed network of single-server queues G(M,K) can be computed

iteratively using the following recursive relation:

with boundary conditions:

1 2 1

1 2 ( ) 1

8-11 Iterative Algorithm (proof)

1 2 1

8-12 Iterative Algorithm – Example

λ

4

λ 3

r =

53

1 2

8-13 Iterative Algorithm – Example

(1,2) (1,1) (0,2) 1 1 2 (1,3) (1,2) (0,3) 2 2 4 (2,2) (2,1) (1,2) 1 2 3

8-14 Marginal Distribution

„ Proposition 1: In a closed Jackson network with M customers, the probability that at steady-state, the number of customers in station j greater than or equal to m is:

1 ( )

1

1 0

′ + ≥

′

′ + + + + = −

8-15 Marginal Distribution

„ Proposition 2: In a closed Jackson network with M customers, the

probability that in steady state there are m customers at station j is:

8-16 Average Throughput

„ Proposition 4: In a closed Jackson network with M customers, the

average throughput of queue j is:

„ Proof 4: Average throughput is the average rate at which customers are serviced in the queue For a single-server queue the service rate is µjwhen there are one or more customers in the queue, and 0 when the queue is empty Thus:

8-17 Example: /M/1 Queues in Tandem

M

K

2 1

8-18 Example: /M/1 Queues in Tandem (cont.)

8-19 Arrival Theorem for Closed Networks

„ Theorem: In a closed Jackson network with M customers, the

occupancy distribution seen by a customer upon arrival at queue j is the

same as the occupancy distribution in a closed network with the arriving customer removed

„ Corollary: In a closed network with M customers, the expected number

of customers found upon arrival by a customer at queue j is equal to the average number of customers at queue j, when the total number of

customers in the closed network is M-1.

„ Intuition: an arriving customer sees the system at a state that does not include itself

8-20 Arrival Theorem (proof)

„ x t( ) ( ( ), , = x t1 x t K( ))state of the network at time t

moving from node i to node j finds the network at state n

1 1

( ) 0 1

1

{ ( ) , ( )} { ( ) } { ( ) | ( ) } ( ) { ( ) | ( )}

{ ( )} { ( ) } { ( ) | ( ) }

( ) ( )

1 1

1

1 1

i K

i K i

′+ >

′

′ + + + + = −

8-21 Mean-Value Analysis

„ Closed network with M customers; performance measures

„ N j (M): average number of customers in queue j

„ T j (M): average a customer spends (per visit) in queue j

„ γj (M): average throughput of queue j

„ Mean-Value Analysis: Calculates N j (M) and T j (M) directly, without first computing G(M) or deriving the stationary distribution of the network

j

j j

i i i

8-22 Mean Value Analysis (proof)

„ Arrival Theorem → expected number of customers that an arrival finds

at queue j is N j (m-1) Service rate for all customer at the queue µ j

„ λ1,…,λK: visit ratios – a solution to flow conservation equations

„ Actual throughput of queue j:

„ Using Little’s Theorem:

„ Summing for all j and noting that ∑ j N j (m) = m:

8-23 Example: /M/1 Queues in Tandem

M

K

2 1

j i

8-24 State-Dependent Service Rates

„ Theorem: The stationary distribution of a closed Jackson network where the nodes have state-dependent service rates is

„ where the normalization constant G(M) is a function of M, the fixed

number of customers in the network