Part II: continuous time Markov chain (CTMC)
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1 Part II: continuous time Markov chain (CTMC) Continuous time discrete state Markov process Definition (Markovian property) X(t) is a CTMC, if for any n and any sequence t 1 <t 2 <,,<t n, we have PXt [ ( ) = j Xt ( ) = i, Xt ( ) = i,..., Xt ( ) = i] n n 1 n 1 n 2 n = PXt [ ( ) = j Xt ( ) = i ] n n 1 n 1 1
2 Exponentially distributed sojourn time of CTMC Sojourn time at state i, τ i, i=1,2, S, is exponentially distributed Proved with Markovian property P[ τ > s+ t τ > s] = P[ τ > t] i i i P[ τ > s+ t] = P[ τ > s] P[ τ > t] i i i P[ τ > s+ t, τ > s] = P[ τ > s] P[ τ > t] i i i i dp[ τ i > s + t] = fτ () sp[ τ ] i i > t ds dp[ τ i > t] = fτ (0) ds i P[ τ > t] i i f ( t) = f (0) e τ τ i τ i ln P[ τ > t] = f (0) t i f (0) t τ i f i P[ τ > t] = e τ i (0) t Sojourn time is exponentially distributed! f τi (0) is transition rate of state i 2
3 Basic elements of CTMC State space: S={1,2,,S}, assume it is discrete and finite for simplicity Transition prob. In matrix form If homogeneous, denote p ( s, t) : = P[ X( t) = j X( s) = i], s t P(t) is independent of s (start time) Chapman Kolmogorov equation Pst (,) = PsuPut (, ) (,), s u t ij Pst (,): = pij (,) st Pt ( ): = Pss (, + t) 3
4 Infinitesimal generator (transition rate matrix) Differentiation of C-K equation, let u t Pst (,) t = PstQt (, ) ( ), s t 0 Q(t) is called the infinitesimal generator of transition matrix function P(s,t) since Its elements are Ptt (, + t) I Qt ( ) = lim t t Pst (,) = Qt () t Pst (,) t (u) (,) Q s Pst = e u pii (, tt+ t) 1 pij (, tt+ t) qii ( t) = lim 0 qij ( t) = lim 0, i j t 0 t 0 t t S qij ( t) = 0, for all i Qte= () 0 j= 1 relation between transition rate q ij and probability p ij -q ii is the rate flow out of i, q ij is the rate flow into i 4
5 Infinitesimal generator If homogeneous, denote Q=Q(t) Derivative of transition probability matrix Pt () t = PtQ () With initial condition P(0)=I, we have Pt () = e Qt where Qt 1 1 e I Qt Q t Q t 2! 3! : = Infinitesimal generator (transition rate matrix) Q can generate the transition probability matrix P(t) Does e Qt converge as t goes to infinity? 5
6 State probability π () t is the state probability at time t We have π( t) = π(0) Pt ( ) = π(0) e Qt Derivative operation dπ () t = π(0) e Qt Q= π( tq ) dt Decompose into element dπ () t i dt π () tq π () tq = + i ii j ji j i 6
7 Steady state probability For ergodic Markov chain, we have lim π ( t) = π = lim p ( t) t Balance equation (stationary distr.) dπ () t In matrix i dt j j ij t = πi () tqii+ π j () tqji= 0 j i π Q = 0 π e = 1 Qe = 0 Comparing this with discrete case Limiting distribution 7
8 Birth-death process A special Markov chain Continuous time: state k transits only to its neighbors, k-1 and k+1, k=1,2,3, model for population dynamics basis of fundamental queues state space, S={0,1,2, } at state k (population size) Birth rate, λ k, k=0,1,2, Death rate, μ k, k=1,2,3, 8
9 Infinitesimal generator Element of Q q k,k+1 = λ k, k=0,1,2, q k,k-1 = μ k, k=1,2,3, q k,j = 0, k-j >1 q k,k = - λ k - μ k Chalk writing: tri-diagonal matrix Q 9
10 State transition probability analysis In Δt period, transition probability of state k k k+1: p k,k+1 (Δt)= λ k Δt+o(Δt) k k-1: p k,k-1 (Δt)= μ k Δt+o(Δt) k k: p k,k (Δt)= [1-λ k Δt-o(Δt)][1- μ k Δt-o(Δt)] = 1- λ k Δt-μ k Δt+o(Δt) k others: p k,j (Δt)= o(δt), k-j >1 10
11 State transition equation State transition equation π ( t+ t) = π ( t) p ( t) + π ( t) p ( t) + π ( t) p ( t) + o( t) k k k, k k 1 k 1, k k+ 1 k+ 1, k k-1 k=1,2, Case of k=0 is omitted, for homework k State k k+1 j k-j >1 at time t at time t+ Δt 11
12 State transition equation State transition equation: π ( t+ t) = π () t ( λ + µ ) tπ () t + λ tπ () t + µ tπ () t + o( t) k k k k k k 1 k 1 k+ 1 k+ 1 With derivative form: dπ k () t dt dπ 0() t dt = ( λ + µ ) π () t + λ π () t + µ π (), t k = 1,2,... k k k k 1 k 1 k+ 1 k+ 1 = λπ () t + µπ (), t k = This is the equation of dπ = πq Study the transient behavior of system states 12
13 Easier way to analyze birth-death process State transition rate diagram λ λ 0 1 λ2 λk 1 λk λ k k-1 k k+1 µ 1 µ 2 µ 3 µ k 1 µ k µ k + 1 This is transition rate, not probability If use probability, multiply dt and add self-loop transition Contain the same information as Q matrix 13
14 State transition rate diagram Look at state k based on state transition rate diagram Flow rate into state k I = λ π () t + µ π () t k k 1 k 1 k+ 1 k+ 1 Flow rate out of state k O = ( λ + µ ) π () t k k k k Derivative of state probability dπ k () t dt = I O = λ π () t + µ π () t ( λ + µ ) π () t k k k 1 k 1 k+ 1 k+ 1 k k k 14
15 State transition rate diagram Look at a state, or a set of neighboring states Change rate of state probability = incoming flow rate outgoing flow rate Balance equation For steady state, i.e., the change rate of state probability is 0, then we have balance equation incoming flow rate = outgoing flow rate, to solve the steady state probability This is commonly used to analyze queuing systems, Markov systems 15
16 Pure birth process Pure birth process Death rate is μ k =0 Birth rates are identical λ k = λ, for simplicity We have the derivative equation of state prob. dπ () t k dt dπ 0() t dt = λπ ( t) λπ ( t), k = 1, 2,... k 1 = λπ ( t), k = 0 0 k assume initial state is at k=0, i.e., π 0 (0)=1, we have π () t 0 t = e λ 16
17 Pure birth process For state k=1 dπ1() t dt we have = λπ () t + Generally, we have 1 λt λe π t = λte λ 1 () t k ( λt) λt π k ( t) = e, t 0, k = 0,1, 2,... k! The same as Poisson distribution! 17
18 Pure birth process and Poisson process Pure birth process is exactly a Poisson process Exponential inter-arrival time with mean 1/λ k, number of arrivals during [0,t] k ( λt) λt Pk = e, k = 0,1, 2,... k! Z-transform of random variable k To calculate the mean and variance of k Laplace transform of inter-arrival time Show it or exercise on class To calculate the mean and variance of inter-arrival time 18
19 Arrival time of Poisson process X is the time that exactly having k arrivals X = t 1 + t t k t k is the inter-arrival time of the kth arrival X is also called the k-erlang distributed r.v. The probability density function of X Use convolution of Laplace transform k λ () = () = λ + s * * X s A s k Look at table of Laplace transform k 1 ( λx) fx ( x) = λ e ( k 1)! arrival rate k-1 arrival during (0,x) λx 19
20 Elementary queueing system Kendall notation Little s Law PASTA theorem Basics of M/M/1 queue 20
21 History of queueing theory An old research area Started in 1909, by Agner Erlang (to model the Copenhagen telephone exchange in Denmark) Booming after 1950 s, to model computer/communication/ David G. Kendall introduced an A/B/C queueing notation in 1953 used in modern packet switching networks (Internet) in the 1960s by Leonard Kleinrock (UCLA, first host of Internet in 1969, his book for queueing) A branch of operations research Agner Erlang , Denmark 21
22 Model and analysis tool Modeling and analyze many systems Telephony systems, exchanger of telephone lines Computer systems Communication systems Transportation systems Production systems Hospital/banks/ Limitation Too restrictive, some assumptions Use other alternative methods, simulation/tools, 22
23 A general queueing system A general queueing system Customer arrival Customer departure waiting room Service facility A generic model for Machine, computer system, communication system, intersection of roads, etc. 23
24 The basic elements of queueing system Arrival process of customers Interarrival time i.i.d., e.g, Poisson arrival Arrival one by one, or in batches, etc. Behavior of customers Patient or impatient; different type of customers Service time i.i.d., e.g., exponential; load/state dependent; 24
25 The basic elements of queueing Service discipline system (cont.) FCFS: first come first serve, LCFS: last come first serve (stack), RS: randomly serve, priority, SRPT: shortest remaining processing time first, PS: processor sharing Service capacity (number of servers) Single server or a group of servers Waiting room (system capacity) Finite or infinite; buffer size design 25
26 Kendall notation A family of notation symbols for different categories of queues Proposed by David G. Kendall in 1953 Professor of Oxford Univ. ( ) and Cambridge Univ. ( ) David G. Kendall , England 26
27 Kendall notation (cont.) A family of notation symbols for different categories of queues A/B/C/K/N/D A: distribution of interarrival time, M or G or D B: distribution of service time, M or G or D C: number of servers K: system capacity, infinite by default N: number of total customers, infinite by default D: the service discipline For default, K =, N = and D = FCFS 27
28 Example of Kendall notation M/M/1; M/G/1; G/M/1; G/G/1; M/D/1; M/Er/1; M/PH/1; MAP/M/1 M/M/c; M/M/1/B; M/M/ //N M/M/c/K/N/ M/M/c/K//LCFS M/M/1///PS 28
29 A joke Different of type of queues Quote from Internet 29
30 Utilization factor For a queue with single server λ: the arrival rate of customers x : the mean service time Utilization factor (or traffic intensity): Physical meaning: Single server: time fraction that server is busy Multiple server: fraction of busy servers ρ = λx / m Offered load is defined as multiple servers r: = λx ρ: = λx for single or 30
31 Performance measures Distribution of performance measures Distribution of waiting time W and sojourn time T of customers; E{W}, Var{W}, Pr{W>t} Distribution of number of customers in the queue N or in the system N q ; E{N}, Var{N}, Pr{N>n} Distribution of busy period of the server BP; E{BP}, Var{BP}, Pr{BP>t} Mean of performance measures Mean waiting time, mean sojourn time/mean response time; Average number of customers, average queue length Average length of busy period Throughput of the system Open system: equals the arrival rate; Closed system: needs calculation and analysis 31
32 The Little s Law For any stable queueing system L: average number of customers in the system λ: arrival rate, average number of arrivals per unit time T: mean response time/sojourn time of customers John Little, (1928-), professor of MIT L = λt 32
33 The Little s Law (cont.) Apply it to the queue (excluding the server) L q : average number of customers in the queue λ: arrival rate, average number of arrivals per unit time W: mean waiting time of customers Lq = λw 33
34 The Little s Law (cont.) Apply it to the server only ρ: average number of customers in the server, λ: average number of arrivals to the server per unit time : mean service time of customers x ρ = λx For M/M/1, we have ρ = λ µ 34
35 The Little s Law (cont.) Applicability Very general, G/G/c Applicable to any queue which is STABLE Applicable to any subsystem of the queue Limitation Inapplicable to unstable queue Only mean metrics, inapplicable to study the transient metrics or distributions 35
36 PASTA Theorem Poisson Arrivals See Time Averages For queues with Poisson arrivals, M/./. The arriving customers find the same mean measures as that observed by an outside observer at an arbitrary point of time Intuitively explained by the fact that Poisson arrivals occurs completely random in time (purely random sampling) Analog: sampling theorem in signal processing? 36
37 PASTA Theorem (cont.) Applicable to any queues with Poisson arrival M/M/1, M/G/1, etc. Not valid for some queues, e.g., D/D/1 Empty at time 0, arrive at 1,3,5,, service time is 1. Arrivals see empty queue, while average number of customers is 1/2 Little s Law and PASTA theorem is very fundamental and important in queueing theory E.g., Mean Value Analysis (MVA) for queueing networks Calculate the mean performance metrics, L, W, T, etc. 37
38 M/M/1 queue System parameters Poisson arrival rate λ, service rate μ Infinite capacity, FCFS Study the performance metrics Customer arrival Customer departure waiting room Service facility 38
39 Dynamics of M/M/1 queue server queue C 1 C 2 C 3 C 1 C 2 C 3 C t t Departure Start service C 1 C 2 C 3 C 4 Arrival # of customer t 39
40 State transition rate diagram of M/M/1 State transition rate diagram state: the number of customers in the system λ λ λ λ λ λ k-1 k k+1 µ µ µ µ µ µ µ A simple birth-death process 40
41 Equilibrium behavior of M/M/1 When the system reaches steady Global balance equation πλ= πµ 0 1 π ( λ + µ ) = π µ + π λ, = 1,2,... n n+ 1 n 1 n Local balance equation πλ n = π n 1 µ, n = 0,1, 2,... + Normalization equation n= 0 π n = 1 41
42 Equilibrium behavior of M/M/1 Chalk writing (cont.) Derive the steady state distribution 1 n n πn = ρ = (1 ρρ ) G G is called the normalization constant, G=1/(1-ρ) 3 more ways to solve the local balance equation recursion, Z-transform, direct approach of solving difference equation Very simple to solve the local balance equation 42
43 Key performance metrics of M/M/1 (time average metrics) Average number of customers in the system, L We have Variance is ρ/(1-ρ) 2 Pr{n k}: Average queue length (excluding the customer being served) We have Chalk writing n n= 0 n= 0 The curve of L w.r.t. ρ n L nπ n(1 ρρ ) ρ 1 ρ = = = k Pr{ n k} = π = ρ i= k 2 ρ Lq = ( n 1) πn = L ρ = 1 ρ n= 1 i Trick: use dev. to avoid integration by parts! 43
44 Key performance metrics of M/M/1 (customer average metrics) Mean response time of customers T By Little s law, we have Mean waiting time of customers (excluding the service time) We have Chalk writing The curve of T, W w.r.t. ρ T = L/ λ = 1 µ λ 1 λ ρ W = T = = = ρt µ µ ( µ λ ) µ λ Example, double λ and μ, how are L,T,W changing? 44
45 Another way to calculate mean performance metrics (MVA) Mean Value Analysis (MVA) Combine the Little s law and PASTA theorem No need to know the distribution of steady state Analysis process Seen by an arriving customer, mean response time is (should be equivalent to T by PASTA) T=L/μ+1/μ Called arrival relation Little s law: L=λT Combine to obtain: T= 1/(μ- λ), L= ρ/(1- ρ), etc. 45
46 Distribution of sojourn time We focus on an arriving customer S: the sojourn time of the arriving customer L q : # of customers in the system seen by the arrival B k : service time of the kth customer, k=1,, L q We have S a L + 1 = k = 1 B k Since B k and L q are independent, we further have a L + 1 n+ 1 k k k= 1 n= 0 k= 1 a P( S > t) = P B > t = P B > t P( L = n) 46
47 Distribution of sojourn time (cont.) By PASTA theorem, n+ 1 P Bk > t is a n+1 stage Erlang distribution So, k = 1 a n PL ( = n) = π = (1 ρρ ) n A trick to transfer cdf of Erlang distr. to sum of pmf of Poisson distr. Weighted sum of Erlang distribution could be exp. distr. Weighted sum of exponential distr. could be?? The sojourn time is exponentially distributed with parameter μ(1-ρ). Interesting! 47
48 Distribution of sojourn time (cont.) Another easy way is using Laplace transform Since we have The above is an exponential distribution with parameter μ(1-ρ). (1 ) t PS ( > t) = e µ ρ 48
49 Distribution of waiting time, W Since S=W+B, so We have W=0 with probability 1- ρ; W is exponentially distributed with parameter μ(1- ρ) with probability ρ. Very Special! one part is pmf, the other part is pdf 49
50 Busy period # of customer t Busy Idle period period IP(idle period) is exponentially distributed with parameter λ E( BP) E( BP) Mean busy period: = = 1 µ λ So, it equals T. E( BP) = EBP ( ) + EIP ( ) EBP ( ) + 1/ λ ρ 50
51 Distribution of busy period It is complicated use Laplace-transform and recursion analysis Omitted for simplicity pdf of BP: Where I 1 (.) is the modified Bessel function of the first kind of order 1, 51
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