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Job Poole
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8 l t t L = t= l t l t t F t t S t,i i t t l t = F t i= S t,i, L = F t S t,i, t= i= F t t S t,i t i E t t t e t,i

9 e, e 3, e 3,2 e 4, e 6, x e i =,..., E t t L t,i F t,i l l =,..., L t,i e t,i l t + l S t,i t,i l,j j =,..., F j t + l l L = E t t= i= min{l t,i, t} l= l F t,i j= S t,i l,j, E t L t,i F t,i l S t,i l,j t i l j e 3,2 L 3,2 = 2 F 3,2 = 2 F 3,2 2 = e 3,2 S 3,2, S 3,2,2

10 S 3,2 2, L = L + L L = (L t ) t=,2,... L r VaR α (L) = VaR α (L ) + + r (VaR α(l 2 ) VaR α (L )) ( + r) (VaR α(l ) VaR α (L )) ( ) r = ( + r) VaR r α(l t t ) + ( + r) VaR α(l ) t= ES α (L) = ES α (L ) + + r (ES α(l 2 ) ES α (L )) ( + r) (ES α(l ) ES α (L )) ( ) r = ( + r) ES r α(l t t ) + ( + r) ES α(l ) t= t =,..., ES α (L t ) ES α (L ) ES α (L ) ES α (L 2 ) ES α (L 2 ) ES α (L ) ES α (L 3 ) ES α (L 2 )

12 N

13 F S E L t F t E t (N t, /n)

14 L F t r =.4373, p =.4 N = 374, p =.2344 S t,i µ = , λ =.787 k = 2.39, σ =.783, θ =. F t r =.4473, p =.5 N = 295, p =.2284 S t,i µ = 59.2, λ =.745 k =.977, σ =.5854, θ =. E t r =.7443, p =.458 λ = 5.56 L t,i r =.538, p =.4746 λ =.727 F t,i l λ =.8923 N = 3, p =.2974 S t,i l,j µ = 3.268, λ =.869 µ = 2.885, σ = 2.49

15 L F t r =.663, p =.4 λ = S t,i µ = , λ = k =.2662, σ = 9.52, θ =. F t r =.6982, p =.48 λ = S t,i µ = 4.559, λ = k =.975, σ = , θ =. E t r =.722, p =.8 λ = 8.75 L t,i r =.869, p =.2447 λ = F t,i l λ =.76 N = 4, p =.765 S t,i l,j µ = 22.9, λ = 3.65 µ = 3.677, σ = 2.76

16 N t F t E t n n = 6397 n = 3897 /n α = 99.9%

17 L L L r = 5%

18 r = 5%

19 L = F t S t,i, t= iid iid S t,i (µ, λ) F t (N t, ) N iid n t (r, p) ψ Ft F t ( ) r p ψ Ft (u) = ψ Nt (ψ Ft N t (u)) = p(( ) + u), n n i= ψ Ft (u) φ L t L φ L (u) = ( ( ψ Ft φst,i (u) )), ( ( )) λ φ St,i (u) = exp 2µ2 iu S µ λ t,i [ (L ) k] [ ] = i k d k (u) du k φ L u= µ(l ) = µpr n ( p) σ 2 (L ) = µ3 pr λn ( p) + µ2 pr (n + p np) n 2 ( p) 2 (L ) = pr σ 3 (L ) (L ) = pr σ 4 (L ) ( 3µ 5 + 3µ4 (n+p np) λ 2 n( p) λn 2 ( p) 2 ) + µ3 (n+p np)(n+2p np) n 3 ( p) 3 ( 6µ 5 p 2 ( r+2) 3µ5 (2λ 2 +5λµ+5µ 2 ) + µ4 (n+p np) λn 3 ( p) 3 λ 3 n( p) n 2 ( p) 2 ) 3µ5 p(6λ+5µ+2 λr+ µr) + 3µ4 p( r+2)(n+p np) 2 λ 2 n 2 ( p) 2 n 4 ( p) 4 VaR α (L ) = µ(l ) + σ(l ) Φ (α)

20 ES α (L ) = µ(l ) + σ(l ) ϕ(φ (α)) α, Φ ( ) ϕ( ) Z Y = a + b ln ( ) Z c d a b c d Z ( ( )) Φ VaR α (L ) = µ(l ) + σ(l (α) a ) c + d exp b ES α K = a + b ln (L ) = µ(l ) + σ(l ) c Φ( α)+d exp( ( VaR α (L ) µ(l ) σ(l ) ) c b ln(d) 2b 2 a b ) Φ( (K b )), α S t,i iid l,j L = (µ, λ) F t,i l t= min{l t,i, t} l= E t F t,i i= l j= S t,i l,j, iid ( λ) E t iid (N t, t,i iid ) F n l (r, p ) t,i iid L (r, p ) L

21 r = 5%

22 E t E t L t,i min{l t,i, t} P (L t,i = x) x =,..., t L A ES N n= AES n = ES(L) L ES( A ES n m m = lim L n + hl m ) ES( n m L n), h h L n n

28 L L L r = %

29 L L L r = %

BRANCHING PROCESSES AND THEIR APPLICATIONS: Lecture 15: Crump-Mode-Jagers processes and queueing systems with processor sharing

BRANCHING PROCESSES AND THEIR APPLICATIONS: Lecture 5: Crump-Mode-Jagers processes and queueing systems with processor sharing June 7, 5 Crump-Mode-Jagers process counted by random characteristics We give