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Queueing Theory Outline • Mean value analysis for Jackson networks – Arrival theorem • Cyclic network • Extension of Jackson networks • BCMP network • More than classical queueing theory Li Xia, Tsinghua Univ. 1 Mean value analysis • Two methods to analyze closed Jackson network – Buzen’s algorithm to compute G(N) and distribution – Mean value analysis to recursively compute the average performance metrics; also can recursively compute marginal distribution • Mean value analysis, proposed by 1. Reiser, M.; Lavenberg, S. S. (1980). "Mean-Value Analysis of Closed Multichain Queuing Networks". Journal of the ACM 27(2): 313-322. (IBM Zurich research, IBM Watson research) 2. Sevcik, K. C.; Mitrani, I. (1981). "The Distribution of Queuing Network States at Input and Output Instants". Journal of the ACM 28 (2): 358-371. Li Xia, Tsinghua Univ. 2 Arrival theorem • Arrival theorem – A general case of PASTA theorem – Also called random observer property (ROP) or job observer property – “upon arrival at a station, a job observes the system as if in steady state at an arbitrary instant for the system without that job” Li Xia, Tsinghua Univ. 3 Arrival theorem • Applicability condition – always hold in open product-form networks with unbounded queues at each node (not Poisson arrival) – may not hold for some networks • Cyclic queue with M=2 and N=2, D/D/1, μ=1 and each server starts with 1 job, # of jobs seen by job 1 is 0, not equals 0.5 – for Poisson arrival, PASTA Theorem – for open Jackson network • q(n)=p(n) – for closed Jackson network • qi(N,n-i)=p(N-1,n-i), the statistics seen by an arrival customer is equal to that of the steady network with one customer less. Li Xia, Tsinghua Univ. 4 Mean value analysis • Based on two basic principles – Arrival theorem (PASTA or ROP for Jackson networks) – Little’s law • For closed Jackson network – qn (N)=pn(N-1), queue length seen by arrival equals that of the network with one less customer – Little’s law is applicable throughout the network Li Xia, Tsinghua Univ. 5 Mean value analysis • With Arrival theorem – Wi(N) = [1+Li(N-1)]/μi (also valid for M/M/1 or M/M/c) • Wi(N): mean response time at node i for a network with N customers • Li(N-1): average number of customers at node i for a network with N-1 customers • With Little’s law – Li(N) =λi(N)Wi(N) • λi(N): throughput (arrival rate) of node i in an N- customer network, which is unknown Li Xia, Tsinghua Univ. 6 Calculation of λi(n) • Calculate visit ratio vi by traffic equations M Add one more equation: v v r, for all i1,..., M i j ji v1+v2+…+vM = 1 j1 – vi is the relative throughput of node i n • Calculate λ(n): ()n M vW() n i1 ii • Throughput of node i: λi(n)=λ(n)vi Li Xia, Tsinghua Univ. 7 Algorithm of mean value analysis • Solve traffic equations to obtain vi, i=1,2,…,M • Initialize Li(0)=0, i=1,2,…,M • For n=1:N, calculate – Wi(n) = [1+Li(n-1)]/μi – λ(n)=n/[v1W1(n)+…+vMWM(n)] – λi(n)= λ(n) vi, i=1,2,…,M – Li(n) =λi(n)Wi(n), i=1,2,…,M Li Xia, Tsinghua Univ. 8 Discussion of mean value analysis • Calculate the average metrics easily – Average queue length, mean waiting time, mean response time – Also calculate the marginal distribution recursively • Recursive algorithm – Complexity is linear to the system size • For multiclass networks – Also applicable, but complexity grows exponentially with the number of classes Li Xia, Tsinghua Univ. 9 Example • Similar to Example 4.5 in page 203 of Gross’ book (machine repair problem) – Closed Jackson network with M=3, N=2 – Exception: all the nodes are single-server – Service rate: μ1=2, μ2=1, μ3=3 – Routing prob.: r12=3/4, r13=1/4, r21=2/3, r23=1/3, r31=1 Li Xia, Tsinghua Univ. 10 Example (cont.) 2 v v v • Visit rate equation set: 13 2 3 3 vv Let v1=1, solve the equation 214 11 v =3/4, v =1/2 v3 v 1 v 2 2 3 43 • State space: (M+N-1) choose N, it is 6 (0,0,2), (0,1,1), (0,2,0), (1,0,1), (1,1,0), (2,0,0) • Buzen’s algorithm to compute G(N), then compute the steady state distribution π = … Li Xia, Tsinghua Univ. 11 Example (cont.) • MVA method – For n=1: • W1(1)=[1+L1(0)]/μ1=1/2; W2(1)=1; W3(1)=1/3 • λ(1)=1/[1*1/2+3/4*1+1/2*1/3]=12/17 • λ1(1)= λ(1)v1=12/17; λ2(1)=9/17; λ3(1)=6/17 • L1(1)= λ1(1)W1(1)=6/17; L2(1)=9/17; L3(1)=2/17 – For n=2: • W1(2)=[1+L1(1)]/μ1=23/34; W2(2)=26/17; W3(2)=19/51 • λ(2)=2/[1*23/34+3/4*26/17+1/2*19/51]=204/205 • λ1(2)= λ(2)v1=204/205; λ2(2)=153/205; λ3(2)=102/205 • L1(2)= λ1(2)W1(2)=138/205; L2(2)=234/205; L3(2)=38/205 Li Xia, Tsinghua Univ. 12 Thinking of closed Jackson network • Relation of parameters – Arrival rate (throughput) λ v.s. service rates μ • μ increases, λ increases – Arrival rate λ v.s. number of customers N • N increases, λ increases with a upper bound • MVA for marginal distribution – Recursive formula, similar to M/M/1 pi(n,N)=λi(N)/μi*pi(n-1,N-1) Given marginal distribution Compute steady state distribution by Jackson theorem, as if independent , multiply … Li Xia, Tsinghua Univ. 13 Cyclic network • A special case of closed Jackson network – rij=1, if j=i+1 and 0<i<M; rM1=1; otherwise rij=0 μ1 μ5 μ2 μ4 μ3 Li Xia, Tsinghua Univ. 14 Cyclic network • Product-form solution of steady state distribution – Traffic equation is special, vi+1=vi, so set all vi=1 – ρi= vi/μi=1/ μi 11n12 n nM pnM12 n n n 12 M GNGN()()12 M 1 GN() where n12 n nM n1 ... nM N 12 M Li Xia, Tsinghua Univ. 15 Extension of Jackson networks • Load-dependent arrival rate and service rate – Similar results, product form solution • Consider travel time between nodes – Model the travel time as extra nodes with ample servers, still keep the form of Jackson networks • Multiclass Jackson network – Each class has its own routing structure, arrival rates and service rates – Applicable for computer, communication systems – BCMP network, still have product-form solution Li Xia, Tsinghua Univ. 16 Non-Jackson network • Many variants from Jackson networks • State-dependent routing probability – Customer has flexibility to decide its next stop • E.g., choose the node with less congestion – Even exponential interarrival and service time, no product-form solution – Use Markov model to do analysis, but suffer from “curse of dimensionality” • Product-form solution avoids this curse of computation • Storage is a curse if need to store every state distribution – Avoid to store every distribution, use iterative calculation, e.g., for all s: L=L+n*p(s), only one iterative variable L Li Xia, Tsinghua Univ. 17 BCMP network • BCMP network definition – M servers, K classes of customers – 4 kinds of service disciplines • FCFS, PS, IS(infinite servers, or ample servers), LCFS with preemptive-resume – Class transition • class k customer from server i transits to server j as class r, with probability qij,kr – Service time distribution • FCFS: IID exponential for all classes; • PS, IS, LCFS: any COX distribution (including exponential) Li Xia, Tsinghua Univ. 18 BCMP network • Steady state distribution of BMCP network has a product-form solution – Handle each server independently and multiply them together – Calculate the normalization constant, • Does exist similar algorithm to Buzen’s? – Scalability, avoid the curse of dimensionality • Applicable to large-scale problems Li Xia, Tsinghua Univ. 19 More than classical queueing theory • Heavy tail traffic • Phase-Type (PH) distribution • Matrix Analytic Method (MAM) Li Xia, Tsinghua Univ. 20 Heavy tail traffic • Assumption – Service time is exponentially distributed – Or: job size is exponentially distributed • In practice, especially in computer system – Job size is not exponentially distributed – Heavy tail, high variance, decreasing failure rate Li Xia, Tsinghua Univ. 21 Heavy tail traffic • Data Measure is important (ACM SIGMETRICS) – Collect the job size in computer system P{jobsize >x} 1 heavy tail exponential 1/2 1/4 1/8 x 1 2 4 8 16 32 Looks like an exponential distribution, F() x ex But actually it is not, 1st moment, 2nd moment, … Li Xia, Tsinghua Univ. 22 Pareto distribution • If we use log-log plot P{jobsize >x} 1 Pareto 1/2 1/4 1/8 Exponential 1/16 x 1 2 4 8 16 32 This fits well a Pareto distribution: F( x ) x , x 1, 0 2 [0.8,1.2] Li Xia, Tsinghua Univ. 23 Pareto distribution • Property of Pareto distribution – Decreasing failure rate f() x x 1 r( x ) , x 1 Fx() xx • The older a job is, the longer it will take CPU time in future – Infinite or near infinite variance • If α≤1: E[x]=∞, E[xn]=∞, E[x|x>a]=∞ • If α>1: E[x]<∞, E[xn]=∞, E[x|x>a]<∞ – Heavy-tail property • 1% largest job comprise 90% system load • More biased than the 80-20 rule (Pareto principle) Li Xia, Tsinghua Univ. 24 Pareto distribution • Also known as Power-law distribution – Hot concept: power-law; small-world – Widely exist in practice, almost everywhere! • Most of the resources/contributions belongs to a few people/units • In business, 80-20 rule • In computer network, heavy-tail traffic Win a lot of awards, top paper, nature/sci.
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