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Renewal processes

Renewal processes. Interarrival times. { 0,T 1 ,T 2 ,..} is an i.i.d. sequence with a common distribution fct. F S i =  j=1 i T j { S i } is a nondecreasing, positive sequence of reneval times (point) The distribution of S i is F (i) F (i ) = f (i ) * F = F * f (i )

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Renewal processes

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  1. Renewal processes

  2. Interarrival times • {0,T1,T2,..} is an i.i.d. sequence with a common distribution fct. F • Si = j=1iTj • {Si} is a nondecreasing, positive sequence of reneval times (point) • The distribution of Si is F(i) • F(i) = f(i) * F = F * f(i) • f(i) is the i-fold convolution of f (f = d/dx F)

  3. The counting process • N(t) = maxi {Si· t} • N(t) counts the number of renewal point before t • M(t) = E[N(t)] is the expected number of renevals before t • M(t) = n n P(N(t)=n) • P(N(t)=n)=P(Sn· t and Sn+1 > t) = P(Sn+1 > t) - P(Sn > t and Sn+1 > t)

  4. The counting process • P(N(t)=n)=P(Sn· t and Sn+1 > t) = P(Sn+1 > t) - P(Sn > t and Sn+1 > t) • Now Sn > t => Sn+1 > t so • P(Sn > t and Sn+1 > t) = P(Sn > t) • Altogether • P(Sn· t and Sn+1 > t) = P(Sn+1 > t) - P(Sn > t) = P(Sn· t) - P(Sn+1· t) = F(n)(t) – F(n+1)(t)

  5. The counting process • M(t) = n n P(N(t)=n) =n n P(Sn· t and Sn+1 > t) = n n (F(n)(t) – F(n+1)(t)) =F(1)(t) + n=2 n F(n)(t) – (n-1)F(n)(t) =F(1)(t) + n=2F(n)(t) =F(t) + n=2f(n) * F =F(t) + f * n=1f(n) * F =F(t) + (f * M)(t)

  6. The renewal density • m(t)= d/dt M(t): is called the renewal density • M(t+h)-M(t) is the expected number of renewals in [t,t+h] • When h is small P(N(t+h)-N(t)>1)=O(h2) and P(N(t+h)-N(t)=1)=O(h) • Thus (M(t+h)-M(t)) ¼ P(N(t+h)-N(t)=1) ¼ h ¢ m(t) • h¢ m(t) approximates the probability of a renewal within [t,t+h]

  7. The renewal density • m(t)= d/dt M(t) • M(t)=F(t) + (f * M)(t) • m(t) = f(t) + d/dt (F * M)(t) • = f(t) + d/dt(s0t M(t-u) f(u) du) • = f(t) + s0t m(t-u) f(u) du • = f(t) + (f * m)(t)

  8. Recurrence times • Backward recurrence time (age): A(t) = t – SN(t) • Forward recurrence time (excess): Y(t) = SN(t)+1 –t • FA,t(a) = P(A(t) · a) • FY,t(y) = P(Y(t) · y)

  9. Distribution of age F A,t(a) = P(A(t) · a) = P(t-S N(t)· a) • We condition on the first renewal, i.e. P(A(t) · a) = s01 P(A(t) · a | S1=s) f(s) ds =s0t P(A(t) · a | S1=s) f(s) ds +st1 P(A(t) · a | S1=s) f(s) ds = s0t P(A(t-s) · a) f(s) ds + st1 P(A(t) · a | S1=s) f(s) ds = s0tFA,t-s(a) f(s) ds + st1 I(t · a) ¢ f(s) ds = s0tFA,t-s(a) f(s) ds + I (t · a} R(t) = (F A,.(a) * f)(t) + I(t · a) R(t)

  10. Distribution of excess • FY,t(y) = P(Y(t) · y) • We condition on the first renewal, i.e. P(Y(t) · y) = s01 P(Y(t) · y | S1=s) f(s) ds =s0t P(Y(t) · y | S1=s) f(s) ds +st1 P(Y(t) · y | S1=s) f(s) ds = s0t P(Y(t-s) · y) f(s) ds + st1 P(Y(t) · y | S1=s) f(s) ds = s0tFY,t-s(y) f(s) ds + st1I(s-t · y)¢ f(s) ds = (FY,.(y) * f)(t) + st1 I(s · (t+y)) ¢ f(s) ds = (FY,.(y) * f)(t) + F(t+y)-F(t)

  11. General solutions • Generally: Z = Q + Z * f • Laplace transform Z(s) = Q(s) + Z(s)f(s) Z(s) (1-f(s)) = Q(s) Z(s) = Q(s) / (1-f(s))

  12. Alternative solution • m(t) = f(t) + (f * m)(t) • Laplace transform m(s) = f(s) + f(s) m(s) m(s) (1-f(s))=f(s) 1-f(s)=f(s)/m(s) • Z(s) (1-f(s)) = Q(s)  Z(s) f(s) = Q(s) m(s) • Z(s) = Q(s) + Z(s) f(s) = Q(s) + Q(s) m(s) • Z(t) = Q(t) + (Q * m)(t)

  13. Example (Poisson) • Poisson process: F(t)=1-exp(-¸ t) f(t) = ¸ exp(-¸ t) R(t) = exp(-¸ t) • m = f + m * f  m(s)=f(s)/(1-f(s)) • f(s) = ¸s exp(-st) exp(-¸ t) dt = ¸ /(s+¸) • m(s) = ¸ /(s+¸)/(1- ¸ /(s+¸)) = ¸ /(s+¸- ¸ )) = ¸ / s • m(t) = ¸ !!!

  14. Limiting renewal densityin general • m(t) = f(t) + (f * m)(t)  • m(s) = f(s) + f(s) m(s)  • m(s)=f(s)/(1-f(s)) • limt -> 1 m(t) = lims -> 0 s m(s) = • lims -> 0 s f(s)/(1-f(s))= (l’Hospital) lims -> 0 d/ds (s f(s)) /lims -> 0 d/ds (1-f(s)) = f(0)/ ((d/ds -f(s))|s=0) = 1/E(Ti) !!!

  15. Example (Poisson) • m(t) = ¸ • FA,t(a) = (FA,.(a) * f)(t) + I(t · a) R(t) • FY,t(y)= (FY,.(y) * f)(t) + F(t+y)-F(t) • Z = Q + Z * f  Z(s) = Q(s) / (1-f(s)) or Z(t) = Q(t) + (Q * m)(t) FA,t(a) = I(t · a) R(t) + ¸s0t I(s · a) R(s) ds = I(t · a) R(t) + ¸s0min(t,a) R(s) ds = I(t · a) exp(-¸ t)+ (1-exp(-min(t,a))) FY,t(y) = (FY,.(y) * f)(t) + F(t+y)-F(t) = F(t+y)-F(t) + ¸s0t F(s+y)-F(s) ds (husk -¸) = exp(-¸ t) - exp(-¸ (t+y)) - exp(-¸ t) + exp(-¸ (t+y)) + (1- exp(-¸ y) ) = 1-exp(-¸ y) = F(y) !!!

  16. Alternating renewal process • Used to model random on/off processes • Network traffic • Power consumption ON OFF Zn Yn Sn-1 Sn Tn = Zn + Yn

  17. Alternating renewal process • I(t) = I(SN(t) < t ·SN(t)+ZN(t)) • I(t) indicates whether t belongs to an on-period. • P(ON at t) = P(I(t)=1)=O(t) • We condition on the first renewal O(t) = P(I(t)=1) = s01 P(I(t)=1 | S1=s) f(s) ds = s0t P(I(t)=1 | S1=s) f(s) ds + st1 P(I(t)=1 | S1=s) f(s) ds = s0t P(I(t-s)=1) f(s) ds + st1 P(t ·Z1 | S1=s) f(s) ds = (O * f)(t) + st1P(Z1¸ t| S1=s) f(s) ds

  18. Alternating renewal process O(t) = (O * f)(t) + st1P(Z1¸ t| S1=s) f(s) ds = (O * f)(t) + s01P(Z1¸ t| S1=s) f(s) ds = (O * f)(t) + P(Z1¸ t) = (O * f)(t) + 1-FZ(t) O(s)=1-FZ(s) + O(s)*f(s)

  19. Example (2 state Markov) • 2 exponential distributions FZ(t)=1-exp(-¸ t) FY(t)=1-exp(-¹ t) f(t) = ¸¹s0t exp(-¸ (t-s)) exp(-¹ s) ds E(T)=E(Y)+E(Z)=1/¹ + 1/¸ limt -> 1=1/E(T)=1/(1/¹+1/¸) • O(s)=1-FZ(s) + O(s)*f(s)  O(s)=(1-FZ(s))/(1-f(s)) or O(s) = 1-FZ(s) + (1-FZ(s)) m(s) • limt -> 1 O(t) = lims -> 0 s O(s) = lims -> 0s(1-FZ(s)) + s(1-FZ(s)) m(s) = lims -> 0(1-FZ(s)) lims -> 0s m(s) = sRZ(t) dt / E(T) • sRZ(t) dt = s 1 ¢RZ(t) dt = tRZ(t) + s t ¢fZ(t) dt -> s t ¢fZ(t) dt = E(Z) • limt -> 1 O(t) = E(Z)/E(T) !!!

  20. Autocorrelation • CII(s) = E((It-E(I))(It+s-E(I))) = E((ItIt+s) – E2(I) • E(I) = limt -> 1 O(t) = E(Z)/E(T) • E(ItIt+s)=P(It and It+s) • Tn = Zn + Yn • SN(t)=SN(t)-1+TN(t) • A(t)=t-SN(t)

  21. Autocorrelation • CII(s) = E((ItIt+s) – E2(I) • t lies in the 1st renewal period • E(I) = E(Z)/E(T) • E(ItIt+s)=P(It and It+s) • P(It and It+s) = s P(t+s · Z1 | S1=x) + P(t ·Z1 and t+s ¸S1 | S1=x) O(t+s-x) f(x) dx

  22. Autocorrelation P(It and It+s) = s P(t+s · Z1 | S1=x) + P(t · Z1 and t+s ¸ S1 | S1=x) O(t+s-x) f(x) dx =s P(t+s · Z1 | S1=x) + P(t · Z1 and t+s ¸ x | S1=x) O(t+s-x) f(x) dx =s P(t+s · Z1 | S1=x) + I(t+s ¸ x)P(t · Z1| S1=x) O(t+s-x) f(x) dx =s P(t+s · z| Z1=z, S1=x) fZ,S(z,x) dzdx + s I(t+s ¸ x)P(t · z| Z1=z, S1=x) O(t+s-x) fZ,S(z,x) dzdx =s I(t+s · z) fS|Z(z,x) fZ(z) dzdx + s I(t+s ¸ x)I(t · z) O(t+s-x) fS|Z(z,x) fZ(z) dzdx =s I(t+s · z) fY(x-z) fZ(z) dzdx + s I(t+s ¸ x)I(t · z) O(t+s-x) fY(x-z) fZ(z) dzdx

  23. Autocorrelation P(It and It+s) = s I(t+s · z) fY(x-z) fZ(z) dzdx + s I(t+s ¸ x)I(t · z) O(t+s-x) fY(x-z) fZ(z) dzdx = sst+sx fY(x-z) fZ(z) dzdx + stt+s stx O(t+s-x) fY(x-z) fZ(z) dzdx = st+s sz fY(x-z) dx fZ(z) dz + stt+s O(t+s-x) stx fY(x-z) fZ(z) dz dx = st+s fZ(z) dx + stt+s O(t+s-x) stx fY(x-z) fZ(z) dz dx = RZ(t+s) + stt+s O(t+s-x) stx fY(x-z) fZ(z) dz dx · RZ(t+s) + sstx fY(x-z) fZ(z) dz dx = RZ(t+s) + P(Z1¸ t) = RZ(t+s)+ RZ(t) \leq 2RZ(2s) (for large s) CII(s) = E((It It+s) – E2(I) \leq E((It It+s) ·2RZ(2s)

  24. Example - Pareto distributions (power/heavy tails) • Let fZ(z)=K z-® I(z ¸z0) ®>1 • FZ(z)= K/(®-1) (z01-® – z1-®) I(z ¸z0) • K= (®-1)/z01-® • FZ(z)= (1 – (z/z0)1-®) I(z ¸z0) • RZ(z)=1-FZ(z) = (z/z0)1-® + I(z ·z0) · (z/z0)1-® CII(s) ¼ 2RZ(2s) = K · (2s)2(H-1) (H - Hurst parameter) H>1/2 : Long Range Dependence (LRD) 1-® = 2(H-1)  H=(1-®)/2+1=3/2-®/2 or ®=3-2H LRD  ® < 2

  25. Sample means • ET = 1/T s0T I(t) dt • I(t) indicates on-state • Var(ET)=E(E T2)= 1/T2 E((s0T I(t) dt)2) = 1/T2 E(s0T I(t) dt s0T I(t) dt) = 1/T2 E(s0Ts0T I(t) I(s) ds dt) = 1/T2s0Ts0T E(I(t) I(s)) ds dt = 1/T2s0Ts0T CII(t-s) ds dt ¼1/T2s0Ts0T 2RZ(2|t-s|) ds dt = 4/T2s0Ts0t RZ(2(t-s)) ds dt

  26. Sample meansfor 2 state Markov process • Var(ET) = 4/T2s0Ts0t RZ(2(t-s)) ds dt • RZ(z) =1-FZ(t)=exp(-¸ t) • Var(ET) ·4/T2s0Ts0t exp(-2¸ (t-s)) ds dt = 4/T2s0T exp(-2¸ t) s0t exp(2¸ s) dx dt = 4/T2/¸s0T exp(-2¸ t) (exp(2¸ t)-1) dt =4/T2/¸s0T (1-exp(-2¸ t)) dt =4/T2/¸ (T+1/2¸ (1-exp(-¸ T)) =4/¸ (1/T+1/2T2¸ (1-exp(-¸ T)) ¼ 4/¸/T

  27. Sample meansfor white noise • w is white noise • B(t)=s0t w(t) dt • B(t) is Brownian motion (Wiener process) • Var(B(t))=´ t (by definition) • ET=1/Ts0t w(t) dt = 1/T B(T) • Var(ET)=1/T2 var(B(T))=1/T2´ T = ´/T • 2 state Markov like white noise

  28. Sample meansfor Brownian motion • B(s)=B(t)+sts w(x) dx = B(t)+b s ¸ t • b and B(t) are independent • CBB(t,s) = E(B(t)B(s)) = E(B(t) (B(t)+b)) =E(B2(t))=´ t = ´ min{t,s} !!! • ET=1/Ts0t B(t) dt • Var(ET)=1/T2s0Ts0tCBB(t,s) ds dt • =1/T2s0Ts0T´ min{t,s} ds dt • =2/T2s0Ts0t´ s ds dt • =1/T2s0T´t2 dt • =1/T2/3´T3 = 1/3 ´ T

  29. Sample meansfor renewal with Pareto distributions • Var(ET) = 4/T2s0Ts0t RZ(t-s) ds dt • RZ(z) = C z1-® • Var(ET) = 4C/T2s0Ts0t (t-s)1-® ds dt = -4C/T2s0Tst0x1-® dx dt = 4C/T2/(2-®) s0Tt2-® dt = 4C/T2/(2-®)/(3-®) T3-® = 4C/(2-®)/(3-®) T1-® For ®¼ 1 right between white noise (s0) and Brownian motion (s-1) fractional Brownian motion (s-1/2) BH(t)=s0t (t-s)H-1/2 w(s) ds

  30. Self similarity • A process X is self similar with Hurst parameter H iff: a-H X(at) is equivalent to X(t) (up to finite joint distributions) CXX(s) = E(X(0)X(s))= (1/s)-2H E(X(0/s)X(s/s)) = s2H CXX(0,1) CXX(t,s)= E(X(t)X(s))= (1/s)-2H E(X(t/s)X(s/s))= = s2H CXX(t/s,1) -> s2H CXX(0,1) for t/s -> 0 CXX(t,t+s)= E(X(t)X(t+s))= E(X(t/(t+s))X((t+s)/(t+s)))= = (t+s)2H CXX(t/(t+s),1) -> (t+s)2H CXX(1,1) for t -> 1

  31. Self similarity • Y(n)=X(n)-X(n-1) • CYY(1,m) = E((X(1)-X(0))(X(1+m)-X(m))) =E(X(1)X(1+m))+E(X(0)X(m))-E(X(1)X(m))-E(X(0)X(1+m)) = m2H (E(X(1/m)X(1/m+1))+E(X(0)X(1))-E(X(1/m)X(1))-E(X(0)X(1/m+1))) = m2H (CXX(1/m,1/m+1)+ CXX(0,1)- CXX(1/m,1)- CXX(0,1/m+1)) = m2H (CXX(1/m,1/m+1) - CXX(0,1/m+1) + CXX(0,1)- CXX(1/m,1)) ¼m2H (CXX(0,1)+1/m D1+1/mD2 + D12/21/m2 + D21/21/m2 + D11/21/m2 + D22/21/m2 -(CXX(0,1)+1/m D2 + D22/21/m2) + CXX(0,1) -(CXX(0,1)+1/m D1+ D11/21/m2 )) = m2H (D12/21/m2 + D21/21/m2) = m2H-2 (D12+ D21)/2

  32. Frequency Domain • RZ(z) = C z1-® • log(RZ(z))=log(C) + (1-®) log(z) • CYY(1,m) = K m2H-2 • SYY(!) = C !1-2H • log(SYY(!))=log(C) + (1-2H) log(!)

  33. Distribution of files sizes

  34. Time averages (aggregated)

  35. Time averages (cont’d)

  36. Aggregated statistics

  37. Estimating the Hurst parameter

  38. Miniproject • Make a statistic on the filesizes of your file system. • Check for power tailed behaviour. • Simulate an M/G/1 queue with power tailed service times. • Compare with results for an M/M/1 queue with the same load: ½ = mean service time/mean interarrival time • Simulate an alternating renewal process with power tailed ”ON” distribution. • Compute an autocorrelation estimate. • Compute estimates of the 1-step increments of sample means. • Compute a power spectrum estimate.

  39. Summary LRD • Let fZ(z)=K z-® I(z ¸ z0) ®>1 • RZ(z) ¼(z/z0)1-® • CII(s) ¼ 2RZ(2s) ·(2s)2(H-1) (H - Hurst parameter, I indicates on period) • H>1/2 : Long Range Dependence (LRD) • LRD  ® < 2 • log(RZ(z))=log(C) + (1-®) log(z)

  40. Summary M/G/1 • M/M/1: Q=½/(1-½) • M/G/1: (Pollachek-Kinchine) Q=½ + (½2 + ¸2 var(S))/2/(1-½) • fS(s)=K s-® I(s ¸s0) ®>1 • E(S2) = K ss01s2s-® ds = K ss01s2-® ds = [s3-®]s01 /(3-®)

  41. Summary SS • A process X is self similar with Hurst parameter H iff: a-H X(at) is equivalent to X(t) (up to finite joint distributions) • Y(n)=X(n)-X(n-1) • CYY(1,m) ¼m2H-2 (D12+ D21)/2 • CYY(1,m) = K m2H-2 • SYY(!) = C !1-2H • log(SYY(!))=log(C) + (1-2H) log(!)

  42. Summary (Sample means) • ET = 1/T s0T I(t) dt • I(t) indicates on-state • Var(ET)= 4/T2s0Ts0t RZ(2(t-s)) ds dt • ¼ 4/¸/T (2 state Markov) • = ´/T (White noise) • = 1/3 ´T (Brownian motion) • = 4C/(2-®)/(3-®) T1-® (Power tail)

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