Networking Theory and Fundamentals - Lecture 8 potx

TCOM 501: Networking Theory & Fundamentals Lecture 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 λ1 µ1 λ2 µ2 r51 λ5 µ5 r53 λ3 „ λ4 µ4 M Closed Network: K nodes with exponential servers „ „ „ µ3 No external arrivals (γi=0) , no departures (ri0=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 „ λ1 Aggregate arrival rates K „ „ λ5 λ3 µ5 µ3 λ4 µ4 Can only be determined up to a constant Use an additional equation to obtain unique solution to the above system, e.g „ Set λ =1, for some node j j „ Set λ =µ , for some node j j j „ Set λ + λ +…+ λ =1 K ni: number of customers at node i Possible states for the closed network n=(n1, n2,…,nK), with ni ≥ and | n |≡ ∑ i =1 ni = M K „ µ2 r53 Relative arrival rates – visit ratios „ λ2 r51 λi = ∑ j =1 λ j rji , i = 1, , K „ µ1 Let F(M) denote the set of all such states M 8-5 Closed Jackson Network „ „ „ Let xi be the number of customers at station i, at steady state Random variables x1, x2,…, xK are not independent – their sum must be equal to M The state x=(x1, x2,…, xK) of the closed network can take values n=(n1, n2,…,nK), with ni ≥ and | n |≡ ∑ i =1 ni = M K „ „ „ 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 p( n ) = P{ x = n} = P{ x1 = n1 ,…, xK = nK } 8-6 Jackson’s Theorem for Closed Networks „ Theorem 1: The stationary distribution of a closed Jackson network is K p( n ) = ρi ni , for all n ∈ F ( M ) = {n : ni ≥ 0, | n |= M } ∏ G ( M ) i =1 „ where the normalization constant G(M) is a function of M G(M) guarantees that {p(n)} is a valid probability distribution ∑ p( n ) = ⇒ G ( M ) = n∈F ( M ) „ „ K ∑ ∏ ρi n i n∈F ( M ) i =1 This stationary distribution is said to have a product-form However: at steady-state the queues are not independent „ {pi(ni)}: marginal stationary distribution of queue i p( n ) ≠ p1 ( n1 ) pK ( nK ) 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: rij* = λ j rji / λi Proof of Theorems & 2: Show that for the corresponding forward and reversed chains p( m)q* ( m, n ) = p( n )q( n, m), n, m ∈ F ( M ), n ≠ m „ Need to prove only for m=Tijn q( n, Tij n ) = µ i rij 1{ni > 0} q* (Tij n, n ) = q* ( n − ei + e j , n ) = µ j rji* 1{ni + > 0} = µ j (λ i rij / λ j ) p(Tij n )q* (Tij n, n ) = p( n )q( n, Tij n ) ⇔ ⇔ p(Tij n )µ j (λi rij / λ j ) = p( n )µ i rij 1{ni > 0} p(Tij n ) = ρ j ρi−1 p( n )1{ni > 0} Verify, exactly as in the open-network case, that: ∑ q( n, m) = ∑ q* ( n, m) = ∑ µi 1{ni > 0}, m≠ n m≠ n i n ∈ F (M ) Closed Jackson Network 8-8 Example: Closed network model for CPU (rate µ1) and I/O (rate µ2) system Upon service completion in 1, customer routed to with probability p2=1-p1, or back to with probability p1 M =fixed number of customers „ „ µ2 Stationary distribution: n customers in and M-n in 1 ρ1M − nρ2n = ρ2n , n = 0,1, , M G( M ) G( M ) Normalization constant − ρ2M +1 G ( M ) = ∑ n =0 ρ = − ρ2 M „ n Utilization factor and throughput of node 1: ρ2M G ( M − 1) U ( M ) = − p(0, M ) = − = G( M ) G( M ) γ1 ( M ) = U ( M )µ1 = µ1 p1 µ1 λ1 = p1λ1 + λ , λ = p2λ1 Choose solution: λ1 = µ1 and λ = p2µ1 pµ ρ1 = 1, ρ2 = µ2 p ( M − n, n ) = „ λ1 G ( M − 1) G( M ) p2 λ2 8-9 Closed Networks: Normalization Constant „ Normalization constant as a function of M and K: G( M , K ) = K ∑ ∏ ρi n i n∈F ( M ) i =1 „ = ∑ n1 + + nK = M ni ≥0 ρ1n1 ρ2n2 ρnKK 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:  M + K − 1   terms in the summation M   „ „ 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: G ( m, k ) = k ∑ ∏ ρi n i = n∈F ( m ) i =1 „ ∑ n1 + + nk = m ρ1n1ρ n22 ρ nkk For a closed network of single-server queues G(M,K) can be computed iteratively using the following recursive relation: G ( m, k ) = G ( m, k − 1) + ρk G ( m − 1, k ) with boundary conditions: G ( m,1) = ρ1m , m = 0,1, , M G (0, k ) = 1, k = 1,2, , K 8-11 Iterative Algorithm (proof) For m > and k > we split the sum into two sums over disjoint sets of states, corresponding to nk = 0, and nk > G ( m, k ) = ∑ ρ1n1 ρ2n2 ρ nkk ∑ ρ1n1 ρ2n2 ρ nkk + n1 + + nk = M = n1 + + nk = m nk = = ∑ n1 + + nk −1 = m ρ1n1 ρ2n2 ∑ n1 + + nk = m nk > ρnk k−−11 + ρ1n1 ρ 2n2 ∑ n1 + + nk = m nk > ρ1n1 ρ 2n2 ρ nkk ρ nkk Note that the first sum is G ( m, k − 1) For the second sum, observing that nk > 0, we define nk = nk′ + 1, where nk′ ≥ Then: ∑ n1 + + nk = m nk > ρ1n1 ρ2n2 ρnkk = ∑ n1 + + nk′ +1= m nk′ ≥ = ρk ρ1n1 ρ2n2 ∑ n1 + + nk′ = m −1 nk′ ≥ Therefore: G ( m, k ) = G ( m, k − 1) + ρk G ( m − 1, k ) ρ1n1 ρ2n2 ρnkk′ +1 ρnkk′ =ρk G ( m − 1, k ) 8-12 Iterative Algorithm – Example „ r51 = r53 = λ1 2µ λ2 2µ λ5 λ3 µ λ4 λ µ M λ1 = λ , λ = λ r51 = r53 ⇒ λ1 = λ λ = λ1 + λ = 2λ1 „ ρ1 = ρ2 = λ1 / 2µ, ρ3 = ρ4 = λ / µ = 2ρ1 ρ5 = λ / λ = 2λ1 / λ = 4ρ1 / ρ, with ρ ≡ λ / µ „ „ „ Visit ratios λi determined up to a multiplicative constant Letting λ1= 2µ, we have: ρ1 = ρ2 = 1, ρ3 = ρ4 = 2, ρ5 = / ρ Calculation of G(M,5) based on the iterative algorithm using these values Iterative Algorithm – Example 8-13 m k „ „ 2 4 11 26 ρ1 = ρ2 = 1, ρ3 = ρ4 = 2, ρ5 = / ρ 23 72 6+4/ρ 23+(6+4/ρ)(4/ρ) 72+[23+(6+4/ρ)(4/ρ)](4/ρ) „ Example: Boundary conditions: G (1,2) = G (1,1) + ρ2G (0,2) = + = G ( m,1) = ρ1m , m = 0,1, , M G (1,3) = G (1,2) + ρ3G (0,3) = + = G (0, k ) = 1, „ 1 1 k = 1,2, , K Iteration: G ( m, k ) = G ( m, k − 1) + ρk G ( m − 1, k ) G (2,2) = G (2,1) + ρ2G (1,2) = + = Marginal Distribution 8-14 „ 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: G ( M − m) , 0≤m≤ M G( M ) n ρ1n1 ρ j j ρnKK Proof 1: P{x j ≥ m} = ∑ p( n ) = ∑ G( M ) n∈F ( M ) n1 +…+ n j +…+ nK = M P{x j ≥ m} = ρ mj „ n j ≥m n j ≥m ∑ = ρ1n1 n1 +…+ n′j + m +…+ nK = M n′j + m ≥ m = ρmj ∑ ρmj G( M ) G( M − m) ρ nKK G( M ) G ( M ) n1 +…+n′j +…+nK = M −m n′j ≥0 = n′ + m ρ jj ρ1n1 n′ ρ jj ρ nKK Marginal Distribution 8-15 „ Proposition 2: In a closed Jackson network with M customers, the probability that in steady state there are m customers at station j is: P{x j = m} = ρ mj „ „ G ( M − m ) − ρi G ( M − m − 1) , 0≤m≤ M G( M ) Proof 2: P{x j = m} = P{x j ≥ m} − P{x j ≥ m + 1} Proposition 3: In a closed Jackson network with M customers, the average number of customers at queue j is: G ( M − m) G( M ) m =1 M M G( M − m) Proof 3: N j = E[ x j ] = ∑ P{x j ≥ m} = ∑ ρ mj G( M ) m =1 m =1 M N j ( M ) = ∑ ρ mj „ 8-16 Average Throughput „ Proposition 4: In a closed Jackson network with M customers, the average throughput of queue j is: γ j (M ) = λ j „ G ( M − 1) G( M ) 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 µj when there are one or more customers in the queue, and when the queue is empty Thus: γ j ( M ) = µ j P{x j ≥ 1} = µ j ⋅ρ j G ( M − 1) G ( M − 1) =λj G( M ) G( M ) Example: /M/1 Queues in Tandem 8-17 µ K µ µ M γ( M ) „ λ1 = λ = … = λ K „ Choose λ1 = λ = … = λ K = µ ⇒ ρi = 1, i = 1, , K „ G( M ) = ∑ n∈F ( M ) ρ ρ n1 n2 ρ nK K  M + K − 1 ( M + K − 1)! = ∑ 1=  = M n∈F ( M )   M !( K − 1)! K 1 M !( K − 1)! , for all n ∈ F ( M ) ρi ni = = „ p( n ) = ∏ M K + − G ( M ) i =1   ( M + K − 1)!   M   8-18 Example: /M/1 Queues in Tandem (cont.) µ K µ µ M γ( M ) „ Average throughput: G ( M − 1) ( M + K − 2)! M !( K − 1)! =µ ⋅ G( M ) ( M − 1)!( K − 1)! ( M + K − 1)! γ( M ) = γ j ( M ) = λ j =µ „ For queue j=1,…,K: N j (M ) = Tj (M ) = „ M M + K −1 M K N j (M ) γ j (M ) = M M + K −1 M + K −1 ⋅ = µ µ K M K Average time-delay: T (M ) = ∑ j Tj (M ) = M + K −1 µ 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 ) = ( x1 (t ), , xK (t )) state of the network at time t „ Tij (t ) probability that a customer moves from station i to j, at time t+ „ For any state n ∈ F ( M ) with ni > , find the conditional probability that a customer moving from node i to node j finds the network at state n P{x (t ) = n, Tij (t )} P{x (t ) = n}P{Tij (t ) | x (t ) = n} αij ( n ) = P{x (t ) = n | Tij (t )} = = P{Tij (t )} ∑ P{x(t ) = m}P{Tij (t ) | x(t ) = m} m∈F ( M ) mi > = p( n )µi rij ∑ m∈F ( M ) mi > „ p( m )µi rij = ρ1n1 ∑ m∈F ( M ) mi > ρini ρ1m1 ρimi ρmKK Changing index mi = mi′ + 1, mi′ ≥ in the sum in the denominator: αij ( n ) = ∑ ρ1n1 ρini ρnKK ρ1m1 ρimi′+1 m1 +…+ mi′ +1+…+ mK = M mi′+1> = ρ1n1 ∑ ρini −1 ρnKK ρ1m1 ρimi′ m1 +…+ mi′ +…+ mK = M −1 mi′≥ „ ρnKK ρmKK ρmKK = ρ1n1 ρini −1 ρ nKK G ( M − 1) This is the probability of state ( n1 ,…, ni − 1,… nK ) in an identical network with M-1 customers ... as in the open-network case, that: ∑ q( n, m) = ∑ q* ( n, m) = ∑ µi 1{ni > 0}, m≠ n m≠ n i n ∈ F (M ) Closed Jackson Network 8- 8 Example: Closed network model for CPU (rate µ1) and I/O (rate... 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- 1 0 Iterative Computation of G(M) „ For any m and k (m=0,…, M; k=1,…, K) define: G... 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 „ λ1 Aggregate arrival rates K „ „ λ5