Priorities (time independent and time dependent) Different service times of different classes at Type-1 nodes -

Transkript

1 E.6 Approximate Analysis of Non-product-Form Queueing Networks Non exponentially distributed service times Priorities (time independent and time dependent) Different service times of different classes at Type-1 nodes - Finite queues (Queueing networks with blocking)) Parallel processing, synchronisation Fork-Join-Systems Queueing networks with heterogeneous M/M/m- oder G/G/m- nodes Simultaneous resource possession E Solution Methods Approximation of a non-product-form queueing network by a product-form queueing network Markovanalysis (expensive) MOSEL Simulation (very expensive) PEPSY Approximation methods PEPSY Extension of the product-form methods Iterative use of product-form methods E.317

2 2 Closed Non-Product-Form Queueing Networks At least one M/G/m-FCFS-node ( -/G/m-FCFS-node) Diffusionsapproximation Maximum entropy method Method of Marie Summation method Bottleneck Approximation Robustness E.318 Robustness: -/G/1-FCFS- and -/G/m-FCFS-nodes are replaced by M/M/1-FCFS- or M/M/m-FCFS-nodes Only little influence of the coefficient of variation of the non-product-form nodes (-/G/1 und -/G/m) in case of closed networks (not valid for open networks!!). This property is called "Robustness". The accuracy of the robustness is sufficient for many applications. E.319

3 Example Network 1 µ 2 p 12 =0.6; p 13 =0.4 µ 1 µ 3 Network 2 µ 1 µ 2 p 12 =0.6; p 13 =0.4 p 21 =0.3; p 23 =0.7 µ 3 Network 3 µ 1 p 11 =0.2; p 12 =0.4; p 13 =0.4; p 21 =0.3; p 22 =0.2; p 23 =0.5; p 31 =0.6; p 32 =0.2; p 31 =0.2; µ 2 µ 3 E.320 Coefficient of variation: Combinations Node 1 Node 2 Node 3 c 2 B 1 c 2 B 2 c 2 B 3 a b c d Service rates: µ 1 µ 2 µ 3 Balanced network Network with bottleneck E.321

4 Throughput of the "Balanced Network" Squared Coefficient Network 1 Network 2 Network 3 of Variations K =5 K =10 K =5 K =10 K =5 K =10 a b c d Throughput of the "Network with Bottleneck": Squared Coefficient Network 1 Network 2 Network 3 of Variations K =5 K =10 K =5 K =10 K =5 K =10 a b c d E.322 Throughput as a function of the number of jobs in the network: K a b c d E.323

5 3 Open Non-Product-Form Queueing Networks Methods: Diffusion approximation Maximum entropy method Decomposition methods Pujolle Whitt Gelenbe Chylla Kühn E.324 Decomposition methods Interarrival times and service times are arbitrarily distributed and given by the first and second moment (Coefficient of variation). Queueing discipline is FCFS, queue length is not limilted Multiple jop classes (no class switching) Multiple Server nodes ( -/G/m ) are possible. E.325

6 Algorithm (Example: method of Whitt): Step 1: Arrival rates and utilization of the nodes: N λ i = λ 0i + λ j p ji j=1 ρ i = λ i m i µ i Step 2: Iterative computation of the coefficient of variation of inter arrival times of the individual nodes (see below). E.326 Step 3: Mean queue length and other performance measures M/M/m-FCFS: Q im/m/m = ρ i 1 ρ i P mi Probability of waiting P mi ( = ρ i für M/M/1) Allen-Cunneen formula for G/G/m-FCFS: Q iac Q im/m/m (c2 Ai + c2 Bi ) 2 E.327

7 Krämer/Langenbach-Belz formula for G/G/m-FCFS: Q iklb Q iac G KLB with: ( 2 e 3 (1 ρ i) (1 ) c2 Ai )2 P mi (c 2 Ai + c2 Bi ), c G KLB = ( 2 Ai 1, (c 2 Ai (1 ρ i ) 1) ) e (c 2 Ai + c2 Bi ), c 2 Ai > 1 E.328 to Step 2: Iterative computation of the coefficient of variation of inter arrival times of the individual nodes ( Whitt): (λ 1i,c 2 1i ) (λ i1,c 2 i1 ) (λ i,c 2 A i ) (λ i,c 2 D i ) (λ Ni,c 2 Ni ) Phase 1: Merging Phase 2: Flow (λ in,c 2 in ) Phase 3: Splitting E.329

8 Initialization: c ij = 1 i, j = 1, 2,..., N Merging: i = 1, 2,..., N c 2 Ai = 1 N c 2 ji λ λ j p ji + c 2 0i λ 0 p 0i i j=1 Flow: i = 1, 2,..., N c 2 Di =1+ρ2 i (c2 Bi 1) mi +(1 ρ 2 i ) (c2 Ai 1) Splitting: i, j = 1, 2,..., N c 2 ij =1+p ij (c 2 Di 1) E.330 Example: Source µ 2 µ 1 µ 4 µ 1 µ 3 Sink p 12 =0.5, p 13 =0.5, p 31 =0.6, p 21 = p 41 =1 µ 1 =12.5, µ 2 = µ 3 =16.666, µ 4 =20 c 2 B 1 =2.0, c 2 B 2 =6.0, c 2 B 3 =0.5, c 2 B 4 =0.2, c 2 04 =4.0 E.331

9 Step 1: Arrival rates and utilizations; λ 1 =20, λ 2 =10, λ 3 =10, λ 4 =4 ρ 1 =0.8, ρ 2 =0.3, ρ 3 =0.6, ρ 4 =0.2 Step 2: Coefficient of variation of inter arrival times: Initialization: c 2 12 = c2 13 = c2 21 = c2 31 = c2 41 =1 E Iteration: Merging: c 2 A 1 = 1 λ 1 (c 2 21 λ 2p 21 + c 2 31 λ 3p 31 + c 41 λ 4 p 41 ) = 1 ( ) = c 2 A 2 =1, c 2 A 3 =1, c 2 A 4 =4 E.333

10 Flow: c 2 D 1 =1+ ρ2 1 (c2 B 1 1) m1 +(1 ρ 2 1 )(c2 A 1 1) = (2 1) 2 +(1 0.64)(1 1) =1.453 c 2 D 2 =1.45, c 2 D 3 =0.82, c 2 D 4 = Splitting: c 2 12 =1+p 12 (c 2 D 1 1) = c 2 13 =1.226, c2 21 =1.450, c2 31 =0.892, c2 41 = E Iteration: Merging: c 2 A 1 =1.762, c 2 A 2 =1.226, c 2 A 3 =1.226, c 2 A 4 =4.0 Flow: c 2 D 1 =1.727, c 2 D 2 =1.656, c 2 D 3 =0.965, c 2 D 4 =3.848 Splitting: c 2 12 =1.363, c2 13 =1.363, c2 21 =1.656 c 2 31 =0.979, c2 41 =3.848 E.335

11 7 Iterationen: Iteration c 2 A 1 c 2 A 2 c 2 A 3 c 2 A E.336 Step 3: Mean number of jobs Iteration c 2 A 1 c 2 A 2 c 2 A 3 c 2 A 4 Methods K 1 K 2 K 3 K 4 AC KLB Sim AC: Allen-Cunneen-Approximation KLB: Krämer/Langenbach-Belz-Approximation E.337

12 4 Priority Queueing Networks Description: Multiple job classes with priorities 1, 2,..., R 1 : highest priority; R : lowest priority Two additional node types: M/M/1-FCFS-NONPRE M/M/1-FCFS- PRE (without preepmtion) (with preemption) Only approximate solutions: Extended MVA (PRIOMVA) Shadow Method E.338 PRIOMVA: MVA for multiple job classes: Mean response time (arrival theorem); [ ] 1 1+ R Type-1,2,4 K is (k 1 r ) µ T ir (k) = ir s=1 (m i =1), 1, Type-3. µ ir (k 1 r )=(k 1,...,k r 1,...,k R ) is the population vector with one class-r joblessinthesystem E.339

13 Mean response time (M/M/1-PRE node): T ir (k) = 1 + r µ ir s=1 1 r 1 K is (k 1 r ) µ is ρ is s=1, Mean response time (M/M/1-NONPRE node): T ir (k) = 1 µ ir + r s=1 K is (k 1 r ) µ is + R 1 r 1 s=r+1 ρ is s=1 ρ is (k 1 r ) µ is, E.340 Shadow Method for M/M/1-FCFS-PRE Nodes: M/M/1-FCFS-PRE-Knoten is replaced by a "shadow node" with R parallel M/M/1-FCFS nodes. Thus a product-form queueing network arises, which can be analyzed by MVA, SCAT or convolution. Since the jobs of the different job classes are processed in parallel, the service times s ir =1/µ ir have to be extended. This has to be done, that the mean response times of the shadow nodes are equal to those of the original M/M/1-FCFS-PRE nodes. This is done iteratively using an approximation formula. The initial value is s ir. E.341

14 Algorithm: Class 1 µ 1 Class i µ i Class R µ R Original Node Shadow Node E.342 Step 1: Transform the original model into the shadow model Step 2: Set λ i,r = 0 Schritt 3: Iteration Step 3.1: Compute the utilization for each shadow node: ρ i,r = λ i,r s i,r Step 3.2: Compute the shadow service times: s i,r = s i,r 1 r 1 ρ i,s s=1 s i,r s i,r : original service time of a class r job at node i job : approximated service time in the shadow node E.343

15 Step 3.3: Compute the new values of the throughput λ ir of class r at node i. Step 4: If the λ ir differ less than ε in two successive iterations, then stop the iteration. Otherwise go back to Step 3.1 E.344