Andrea Marin From Systems to Components: Constructive Methods for Product-Form Solutions: other product-forms Andrea Marin 1 Maria Grazia Vigliotti 2 1 Dipartimento di Informatica Università Ca’ Foscari di Venezia 2 Department of Computing Imperial College London Andrea Marin University of Venice
Andrea Marin Second part: sketch 1 Multiple application of (G)RCAT A class of non-pairwise cooperations are considered. We show how multiple applications of (G)RCAT can still derive the product-form solution when it exists. Case studies: finite capacity queues with skipping [Pittel ’79, Balsamo et al. ’10], G-networks with signals [Harrison ’04b]. 2 Extended Reversed Compound Agent Theorem (ERCAT). The Extended Reversed Compound Agent Theorem [Harrison ’04a] is introduced. Applications for cooperations of pairs of automata which do not yield structural conditions of RCAT are shown. Andrea Marin University of Venice
Introduction Skipping queues G-networks and triggers Andrea Marin Introduction Skipping queues G-networks and triggers Part I Multiple applications of RCAT Andrea Marin University of Venice
Introduction Skipping queues G-networks and triggers Andrea Marin Outline Introduction Skipping queues G-networks and triggers 1 A mild introduction 2 Finite capacity queues with skipping: the RCAT solution 3 Product-form solution for G-networks with positive Andrea Marin University of Venice
Introduction Skipping queues G-networks and triggers Andrea Marin Preliminary Introduction λ 1 λ 1 λ 1 Skipping . . . . . . queues λ N λ N λ N G-networks ... and triggers N 0 N 1 N 2 µ 1 µ 1 µ 1 . . . . . . ( a, µ i ) ( a, µ i ) ( a, µ i ) µ M µ M µ M • Value K a may be interpreted as the sum of the reversed rates of the active transitions labelled by a incoming into each state • In case of Birth and Death processes this may be easily computed, i.e.: � N j = 1 λ j K a = µ i � M j = 1 µ j Andrea Marin University of Venice
Introduction Skipping queues G-networks and triggers Andrea Marin RCAT or GRCAT? Introduction Skipping queues G-networks • The Reversed Compound Agent Theorem (RCAT) and triggers [Harrison ’03] requires each state to have one incoming active transition for each synchronising label. Value K a may be interpreted as the (constant) reversed rate of this unique transition. • The Generalisation (GRCAT) proposed in [Marin et al. ’10] requires each state to have at least one incoming active transition for each synchronising label. Value K a may be interpreted as the (constant) sum of the reversed rates of these transitions. Andrea Marin University of Venice
Introduction Skipping queues G-networks and triggers Andrea Marin Skipping mechanism for queues with finite Introduction capacity Skipping queues G-networks • Consider a tandem of exponential queues, Q 1 and Q 2 and triggers • Q 1 has a finite capacity B 1 > 0 • Customers arrive according to a homogeneous Poisson process at Q 1 • If at the arrival epoch Q 1 is saturated, the customer immediately enters in Q 2 • After service completion in Q 1 customers go to Q 2 Q 1 Q 2 λ µ 2 µ 1 Andrea Marin University of Venice
Introduction Skipping queues G-networks and triggers Andrea Marin Standard RCAT analysis Introduction • Processes: Skipping queues λ λ λ G-networks ( a, λ ) 1 2 0 B 1 Q 1 and triggers ( a, µ 1 ) ( a, µ 1 ) ( a, µ 1 ) ( a, x a ) ( a, x a ) ( a, x a ) 0 Q 2 1 2 µ 2 µ 2 µ 2 • Clearly, the reversed rates of a -transitions are constant, hence K a = λ • Structural (G)RCAT conditions are satisfied • Steady-state distribution: � λ � n 1 � λ � n 2 π ( n 1 , n 2 ) ∝ with 0 ≤ n 1 ≤ B 1 , n 2 ≥ 0 µ 1 µ 2 Andrea Marin University of Venice
Introduction Skipping queues G-networks and triggers Andrea Marin Standard RCAT analysis Introduction • Processes: Skipping queues λ λ λ G-networks ( a, λ ) 1 2 0 B 1 Q 1 and triggers ( a, µ 1 ) ( a, µ 1 ) ( a, µ 1 ) ( a, x a ) ( a, x a ) ( a, x a ) 0 Q 2 1 2 µ 2 µ 2 µ 2 • Clearly, the reversed rates of a -transitions are constant, hence K a = λ • Structural (G)RCAT conditions are satisfied • Steady-state distribution: � λ � n 1 � λ � n 2 π ( n 1 , n 2 ) ∝ with 0 ≤ n 1 ≤ B 1 , n 2 ≥ 0 µ 1 µ 2 Andrea Marin University of Venice
Introduction Skipping queues G-networks and triggers Andrea Marin Standard RCAT analysis Introduction • Processes: Skipping queues λ λ λ G-networks ( a, λ ) 1 2 0 B 1 Q 1 and triggers ( a, µ 1 ) ( a, µ 1 ) ( a, µ 1 ) ( a, x a ) ( a, x a ) ( a, x a ) 0 Q 2 1 2 µ 2 µ 2 µ 2 • Clearly, the reversed rates of a -transitions are constant, hence K a = λ • Structural (G)RCAT conditions are satisfied • Steady-state distribution: � λ � n 1 � λ � n 2 π ( n 1 , n 2 ) ∝ with 0 ≤ n 1 ≤ B 1 , n 2 ≥ 0 µ 1 µ 2 Andrea Marin University of Venice
Introduction Skipping queues G-networks and triggers Andrea Marin Standard RCAT analysis Introduction • Processes: Skipping queues λ λ λ G-networks ( a, λ ) 1 2 0 B 1 Q 1 and triggers ( a, µ 1 ) ( a, µ 1 ) ( a, µ 1 ) ( a, x a ) ( a, x a ) ( a, x a ) 0 Q 2 1 2 µ 2 µ 2 µ 2 • Clearly, the reversed rates of a -transitions are constant, hence K a = λ • Structural (G)RCAT conditions are satisfied • Steady-state distribution: � λ � n 1 � λ � n 2 π ( n 1 , n 2 ) ∝ with 0 ≤ n 1 ≤ B 1 , n 2 ≥ 0 µ 1 µ 2 Andrea Marin University of Venice
Introduction Skipping queues G-networks and triggers Andrea Marin Possible generalisation? Introduction Skipping queues • Consider a sequence of N exponential stations G-networks Q 1 , . . . , Q N with finite capacities B 1 , . . . , B N and triggers • Customers arrive at Q i according to a homogeneous Poisson process with rate λ i , 1 ≤ i ≤ N • At a job completion at queue Q i , the customer tries to enter queue Q i + 1 , 1 ≤ i < N • A customer is allowed to enter Q i if this is not saturated, or must try to enter Q i + 1 otherwise, 1 ≤ i < N • After a job completion at queue Q N or if this is saturated, customers leave the system • Note the system in unconditionally stable Andrea Marin University of Venice
Introduction Skipping queues G-networks and triggers Andrea Marin Are these pairwise cooperations? Introduction Skipping λ 2 λ 3 λ N Q 1 Q 2 Q 3 Q N queues G-networks λ 1 µ 2 µ 3 µ 1 µ N and triggers • Each transition in the system may change the state of only two components but. . . • Consider the cooperation between Q 1 and Q 3 : an arrival or a job completion at Q 1 may generate an arrival at Q 3 depending on the state of Q 2 ! • The cooperation cannot be described only in terms of pairs of queues in isolation • These cases may still be studied by RCAT with multiple applications Andrea Marin University of Venice
Introduction Skipping queues G-networks and triggers Andrea Marin Outline Introduction Skipping queues G-networks and triggers 1 A mild introduction 2 Finite capacity queues with skipping: the RCAT solution 3 Product-form solution for G-networks with positive Andrea Marin University of Venice
Introduction Skipping queues G-networks and triggers Andrea Marin Cooperating processes Introduction λ 1 λ 1 Skipping ( a 12 , λ 1 ) Q 1 0 1 B 1 queues G-networks ( a 12 , µ 1 ) ( a 12 , µ 1 ) ( a 12 , x 12 ) and triggers ( a 23 , K 12 ) λ 2 λ 2 ( a 12 , x 12 ) ( a 12 , x 12 ) Q 2 0 1 B 2 ( a 23 , λ 2 ) ( a 23 , µ 2 ) ( a 23 , µ 2 ) λ 3 λ 3 ( a 23 , x 23 ) ( a 34 , K 23 ) ( a 23 , x 23 ) ( a 23 , x 23 ) ( a 34 , λ 3 ) Q 3 0 1 B 3 ( a 34 , µ 3 ) ( a 34 , µ 3 ) ( a ( N − 1) N , x ( N − 1) N ) ( a ( N − 1) N , x ( N − 1) N ) λ N λ N Q N 0 1 B N λ N µ N µ N Andrea Marin University of Venice
Introduction Skipping queues G-networks and triggers Andrea Marin Peculiarity of the model Introduction Skipping queues G-networks and triggers • For 1 < i < N a self-loop of state B i has two roles : • it is passive with respect to cooperation label a ( i − 1 ) i • it is active with respect to cooperation label a i ( i + 1 ) and has K ( i − 1 ) i as a forward rate • We apply (G)RCAT multiple times adding at each time a new queue Andrea Marin University of Venice
Introduction Skipping queues G-networks and triggers Andrea Marin Application of RCAT to the first two queues Introduction Skipping λ 1 λ 1 queues G-networks ( a 12 , λ 1 ) Q 1 0 1 B 1 and triggers ( a 12 , µ 1 ) ( a 12 , µ 1 ) λ 2 λ 2 ( a 12 , x 12 ) ( a 12 , x 12 ) ( a 12 , x 12 ) ( a 23 , λ 2 ) Q 2 0 1 B 2 ( a 23 , µ 2 ) ( a 23 , µ 2 ) RCAT can be applied because: • Structural conditions on passive transitions are satisfied • Structural conditions on active transitions are satisfied • We have K 12 = λ 1 Andrea Marin University of Venice
Introduction Skipping queues G-networks and triggers Andrea Marin Application of RCAT to the first two queues Introduction Skipping λ 1 λ 1 queues G-networks ( a 12 , λ 1 ) Q 1 0 1 B 1 and triggers ( a 12 , µ 1 ) ( a 12 , µ 1 ) λ 2 λ 2 ( a 12 , x 12 ) ( a 12 , x 12 ) ( a 12 , x 12 ) ( a 23 , λ 2 ) Q 2 0 1 B 2 ( a 23 , µ 2 ) ( a 23 , µ 2 ) RCAT can be applied because: • Structural conditions on passive transitions are satisfied • Structural conditions on active transitions are satisfied • We have K 12 = λ 1 Andrea Marin University of Venice
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