Minimization of cost for given throughput. Maximization of throughput with cost constraint

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1 Optmzaton problems: Mnmzaton of cost for gven throughput Maxmzaton of throughput wth cost constrant Mnmzaton of the mean response tme wth cost constrant Decson varables: System parameters that can be vared to obtan optmal results For computer systems servce rates are often decson varables F.345 Cost functons: Lnear cost functon (c = cost factor for servce rate µ ): C(µ) = c µ =COST Non lnear cost functon: C(µ) = c µ α =COST, α > 1 Cost functon ncludng the costs for man memory C(K): C(µ, K) =C(K)+ c µ α =COST F.346

2 Optmzaton based on Summaton method: Fundamental equaton of the summaton method: f (λ )=K = ρ 1 K 1 K ρ λ, Type-3. µ, Type-1,2,4 (m =1), Smplfed for optmzaton (wthout correcton factor): λe, Type-1, 2, 4, andm =1, µ λe K (λ, µ )=f (λ, µ )= λe, Type-3 IS. µ F.347 System equaton: f (λ, µ )= IS λe µ λe + IS λe µ = K. Results are approxmate (system equaton approxmaton), F.348

3 Maxmzaton of the Throughput λ wth lnear cost functon: Lagrange functon (y 1, y 2 : Lagrange Multpler): ( N ) L(λ, µ 1,...,µ N,y 1,y 2 )=λ + y 1 c µ COST + y 2 λe + µ IS λe IS λe K µ A necessary condton for optmal servce rates µ and maxmum throughput λ s obtaned by dfferentatng wth respect to λ, µ, y 1, y 2 : λ =0, =0, =1,...,N, µ =0, y 1 =0. y 2 F.349 The optmal values are obtaned by solvng the precedng system of equatons: λ = ( N COST K ) 2 c e + K, c e IS ej c j λ j=1 e µ = K +1, e c ej c j λ j=1 e K, e c type IS, type = IS. F.350

4 Mnmzaton of cost subject to a mnmum throughput requrement: Optmal values of the servce rates: ej c j j=1 λe µ = K +1, e ej c j j=1 λe K, e type IS, type = IS. Mnmal cost: C (µ) = µ c. F.351 Mnmzaton of the cost usng a non lnear cost functon: Intalzaton: µ =1, for =1,...,N Iteraton: y = λ ( 1 K λye α c µ α 1 µ = ( λye α c ) 2 α c e µ α 1, ) 1 α 1 + λe, type IS, type = IS. F.352

5 Example: Closed product form queueng network: Maxmzaton of the throughput usng a lnear cost functon K = 20 N = 5 3 M/M/1-FCFS 2 M/G/ IS Vst ratos: e 1 =1, e 2 =0.2, e 3 = e 4 =0.5, e 5 =0.3. Cost factors: c 1 =10, c 2 = c 3 =5, c 4 =2, c 5 =1. Cost constrant: COST = 100 F.353 Results: λ* = 6.189; λ MVA = µ 1 =6.189, µ 2 =1.689, µ 3 =3.812, µ 4 =1.096, µ 5 =1.236, 3 addtonal examples: Example 1 Example 2 Example 3 BFS Complex BFS Complex BFS Complex Method Method Method Method Method Method µ µ µ µ µ λ BFS λ MVA COST F.354

6 Non lnear cost functon: α µ µ µ µ µ λ BFS λ MVA F.355 Optmzaton based on the convoluton algorthm: Maxmzaton of the throughput: Convoluton algorthm and BCMP theorem: λ(µ, K) = G(µ, K 1) G(µ, K) F (k )= ( e µ G(µ, K) = ) k 1 β (k ) NP k =K N F (k ) k!, k m, β (k )= m!m k m 1, k m, 1, m =1. F.356

7 Cost functon: C(µ, K) =C(K)+ c µ α =COST Lagrange functon: ( L(µ, y, K) =λ(µ, K)+y C(K) COST + c µ α ) Dfferentaton: µ =0, =1,...,N, y =0. F.357 We can use the computer algebra program MAPLE to obtan the dervatves and solve the non-lnear system of equatons. Example: Dsk 2 CPU 1 Dsk 3 Dsk 4 Parameter: COST = 500; C(K) = C m K; C m = 5 F.358

8 Cost factors and exponents: c α CPU Dsk Dsk Dsk Results: K λ µ 1 µ 2 µ 3 µ F.359