1 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 Mean-Value Analysis Norton’s Equivalent

8-3 Closed Jackson Networks Closed Network: K nodes with exponential servers No external arrivals (γ i =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 λ j =1, for some node j Set λ j =μ j, for some node j Set λ 1 + λ 2 +…+ λ K =1 n i : number of customers at node i Possible states for the closed network n=(n 1, n 2,…,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 x 1, x 2,…, x K are not independent – their sum must be equal to M The state x=(x 1, x 2,…, x K ) of the closed network can take values n=(n 1, n 2,…,n K ), with Let F(M) denote the set of all such states Define ρ i ≡ λ i /μ i – this is not the actual utilization factor of station i Jackson’s theorem for closed networks gives the stationary distribution

8-6 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 Jackson’s Theorem for Closed Networks

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 μ 1 ) and I/O (rate μ 2 ) system. Upon service completion in 1, customer routed to 2 with probability p 2 =1-p 1, or back to 1 with probability p 1. M =fixed number of customers Stationary distribution: n customers in 2 and M-n in 1 Normalization constant Utilization factor and throughput of node 1:

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

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:

8-11 Iterative Algorithm (proof)

8-12 Iterative Algorithm – Example Visit ratios λ i determined up to a multiplicative constant Letting λ 1 = 2μ, we have: Calculation of G(M,5) based on the iterative algorithm using these values

8-13 Iterative Algorithm – Example Boundary conditions: Iteration: Example:

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: Proof 1:

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: Proof 2: Proposition 3: In a closed Jackson network with M customers, the average number of customers at queue j is: Proof 3:

8-16 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 μ j when there are one or more customers in the queue, and 0 when the queue is empty. Thus: Average Throughput

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

8-18 Example:./M/1 Queues in Tandem (cont.) Average throughput: For queue j=1,…,K: Average time-delay:

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)

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 γ j (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 Iterative calculation: with initial condition: Average throughput

8-22 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. λ 1,…,λ K : 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: Then: Mean Value Analysis (proof)

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

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 Normalization constant: Proof similar to the one for open networks