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i +α 3 1 2 4 5 1 2
ii 3 4 5 6 7 8 9 9.3 2014 6
iii 1 1 2 5 2.1 5 2.2 7 2.3 9 2.4 11 2.5 15 2.6 19 2.7 22 2.8 30 2.9 35 2.10 39 41 3 42 3.1 42 3.2 45 3.3 47 3.4 53 3.5 57 3.6 58 59 4 60 4.1 60 4.2 63 4.3 65 4.4 65
iv 5 70 5.1 70 5.2 83 94 6 95 6.1 95 6.2 98 6.3 101 6.4 104 109 7 110 7.1 M/M/S/S/N 110 7.2 M/M/S/S 117 123 8 124 8.1 M/M/S 124 8.2 M/M/S/K 130 136 9 137 9.1 M/M/1 137 9.2 M/M/1/K 143 9.3 M/G/1 150 171 172 A 174 B 176 C 178 D 179 E 182 F 184
v G 186 H B 191 I C 192 193 211 A α alpha B β beta Γ γ gamma Δ δ delta E ε epsilon Z ζ zeta H η eta Θ θ theta I ι iota K κ kappa Λ λ lambda M μ mu N ν nu Ξ ξ xi O o omicron Π π pi P ρ rho Σ σ sigma T τ tau Υ υ upsilon Φ φ phi X χ chi Ψ ψ psi Ω ω omega
vi 0 0 1 1 1(A) A nc k n k A c A F c (x) X E[X] X I Im[s] s i L max[x 1,...,x n] x 1,...,x n min[x 1,...,x n] x 1,...,x n N(μ, σ 2 ) μ σ 2 O 0 o(δt) Δt 0 Δt 0 np k n k P (A) A P (A, B) A B P (A B) B A Re[s] s A T A U( ) V [X] X Γ ( ) γ( ) δ( ) φ f ( k) f (k) F ˆP! ω A ω A (a, b) [a, b] (a, b] [a, b) ( ) n k f k f k f p ω A ω A a<x<b a x b a<x b a x<b n k nc k A(0,t] a a c B B S B(0,t] C S D(0,t] H H(t) h(t) h K (0,t] { } { } B (0,t] { } C (0,t] { } { } { } H { } H { } E[H]
vii L L q N P p i,j P (t) p i,j (t) Q q i,j q k S t W W (t) w(t) W q W q(t) Wq c(t) Wq c (0) X(t) {X(t)} γ Δt η λ λ k μ μ k π π j π(t) π j (t) ρ { } { } i j P (i, j) t i j t P(t) (i, j) i j Q (i, j) { } { } k { } W W W q Wq t {} k {} { } k { } π j t t j π(t) j 2 A, B F (x) F X (x) X F X,Y (x) X Y F X Y =y (x) Y = y X f(x) f X (x) X f X,Y (x) X Y f X Y =y (x) Y = y X p(x) p X (x) X p X,Y (x) X Y p X Y =y (x) Y = y X S X, Y Ω ω 3 I R T (0,t] (0,t] T c(0,t] (0,t] 4 L L q W n n W W q
viii 5 S i i T i,j i j z f i,j (n) i j n P (n) n p i,j (n) X(n) π(n) π j (n) i j n P (n) (i, j) n n n j π(n) j a i 6 i M {X m(t)} m p m m t n n t n t n {Y (t)} λ m 7 m 9 A A n a k C j J L n L(t) R r(t) T t(0,τ] U j Y Y (t) y(t) α n δ n + n k j n t R (0,τ] t j Y Y n n a s b k b S G S ν π S k
1 1 traffic congestion 1 1.1 A B A B 1 2 5 1 0.01 1
2 1 1.1 A-B 2 1.2 1 50 10 1 0.1 1.2 1 2 20 Agner Krarup Erlang 1
1 3 traffic theory 3 queueing 1 theory 2 4 5 6 1 7 2 8 Paul Baran Donald Watts Davies 1 1 queuing queueing
4 1 3 1.3 1 100 1 Gbps 1 1KB 10 5 1.3 9 1 G giga 10 9 10 bps bit per second 1
5 2 2.1 trialsample sample space Ω event 2.1 Ω Ω = { } S S = {φ, { }, { }, { }} φ S (1) Ω S (2) A S A c = {ω; ω A} S (3) A S B S A B S
6 2 (1) (2) Ω c = φ (A c B c ) c = A B (2.1) (2) (3) A B A B A B A B = φ A B A B exclusive 2.2 3 A B 1 B 2 A = { 3, 6 }, B 1 = { 1, 3, 5 }, B 2 = { 2, 4, 6 } A B 1 = { 1, 3, 5, 6 }, A B 1 = { 3 }, A c = { 1, 2, 4, 5 } B 1 B 2 = φ B 1 B 2 Ω A (1) 0 P (A) 1 (2) P (Ω) =1 ( ) (3) A 1,A 2,A 3,... P A i = i i P (A i ) P (A) A probability (1) (3) 2.3 Ω = { } { } { } P (Ω) =P ( ) =P ( )+P( ) =1 { } { } P ( ) =P ( ) = 1 2
42 3 3.1 3.1 n (n 1) + (n 2) + +2+1= (n 1)n 2 (3.1) 3.2 switching system 3.1 3.2
3.1 43 switching exchange circuit switching store-and-forward switching 3.1.1 3.2 1 3 3.3 3.3 1 3 3 1 3 1 3 3 1 1 3 3 3 2 3
44 3 3.1.2 Internet LAN Local Area Network 3.2 1 3 3.4 3.4 1 PDU Protocol Data Unit protocol 1 3 PDU PDU 2 3 PDU PDU buffer 3.2 router hub 1 PDU 48 3 PDU message packet cell PDU PDU
3.2 45 3.5 LAN Ethernet ( ) header 6 DA Destination Address 6 SA Source Address 2 PT Protocol Type 3 IP Internet Protocol 46 1500 I Information FCS Frame Check Sequence trailer 3.5 PDU PDU 3.4 3.2 1 call 1
46 3 switching system 3.6 source trunk switching element N S N S 3.6 bottleneck 1 holding
3.3 47 holding time service time 1 1 loss system 2 waiting system loss lost call crossbar switch 3.7 3.7 3.3 1 1 1
48 3 (0,t] A(0,t] B(0,t] B B = B(0,t] A(0,t] (3.2) B loss probability h (0,t] T (0,t] T (0,t]=A(0,t] h (3.3) T (0,t] (0,t] offered traffic T (0,t] a a = T (0,t] t = A(0,t] h t (3.4) a offered load (3.4) erlang erl (0,t] T c (0,t] T c (0,t]=(A(0,t] B(0,t]) h (3.5) T c (0,t] (0,t] carried traffic carried load a c a c = T c(0,t] t = (A(0,t] B(0,t]) h t = A(0,t] h t ( 1 B(0,t] ) A(0,t] = a(1 B) (3.6) (3.6) B B = a a c (3.7) a a a c
3.3 49 3.8 a a c 3.8 S (0,t] T c (0,t] St St T c (0,t] η = T c(0,t] St = a c S (3.8) η utilization 3.1 3.9 S =2 60 A(0, 60 ] = 13 [ ] B(0, 60 ] = 1 [ ] B B = B(0, 60 ] A(0, 60 ] = 1 13 0.077 3.9
70 5 5.1 Markov process Markov chain 5.1.1 {X(n)} 2.16 n =0, 1, 2,... 5.1 5.1
5.1 71 0, 1, 2,...,n i 0,i 1,i 2,...,i n n +1 j P (X(n +1)=j X(0) = i 0,X(1) = i 1,...,X(n) =i n ) P (X(n +1)=j X(0) = i 0,X(1) = i 1,...,X(n) =i n ) = P (X(n +1)=j X(n) =i n ) (5.1) Markov property {X(n); n =0, 1, 2,...} i 0,i 1,...,i n j n (5.1) discrete-time Markov chain i j n P (X(n +1)=j X(n) =i) =P (X(1) = j X(0) = i) (5.2) stationary (5.2) p i,j P (X(1) = j X(0) = i) =p i,j (5.3) p i,j i j transition probability 5.2 state transition diagram 0 5.2 p 0,0 =1 α, p 0,1 = α, p 0,2 =0, p 0,3 =0 p 1,0 =0, p 1,1 =0, p 1,2 =1, p 1,3 =0 p 2,0 =0, p 2,1 =0, p 2,2 =0, p 2,3 =1 p 3,0 =0, p 3,1 = β, p 3,2 =1 β, p 3,3 =0 5.2
72 5 5.1 5.2 X(0) initial state i 0 i 1,i 2,...,i n P (X(0) = i 0,X(1) = i 1,...,X(n) =i n ) = P (X(0) = i 0,X(1) = i 1,...,X(n 1) = i n 1 ) P (X(0) = i 0,X(1) = i 1,...,X(n) =i n ) P (X(0) = i 0,X(1) = i 1,...,X(n 1) = i n 1 ) = P (X(0) = i 0,X(1) = i 1,...,X(n 1) = i n 1 ) P (X(n) =i n X(0) = i 0,X(1) = X 1,...,X(n 1) = i n 1 ) (5.4) (5.1) P (X(0) = i 0,X(1) = i 1,...,X(n) =i n ) = P (X(0) = i 0,X(1) = i 1,...,X(n 1) = i n 1 ) P (X(n) =i n X(n 1) = i n 1 ) = P (X(0) = i 0,X(1) = i 1,...,X(n 1) = i n 1 )p in 1,i n (5.5) P (X(0) = i 0,X(1) = i 1,...,X(n) =i n ) = P (X(0) = i 0 )p i0,i 1 p i1,i 2 p in 1,i n (5.6) i S i P (S i k) = k 1 m=0 p m i,i(1 p i,i ) (k =1, 2, 3,...) (5.7) S i S i i sojourn time 5.1.2 2.10
5.1 73 initial distribution p i,j (i, j =0, 1, 2,...) n X(n) j π j (n) P (X(n) =j) =π j (n) (5.8) π j (n) X(n) π j (n) j π(n) [ π 0 (n),π 1 (n),π 2 (n),...]=π(n) (5.9) n 0 i n j 1 p i,j (n) P (X(n) =j X(0) = i) =p i,j (n) (5.10) p i,j (n) i j n (5.10) n =1 (5.3) n =1 (1) p i,j i j m n P (X(m + n) =j X(m) =i) =P (X(n) =j X(0) = i) (5.11) i j n p i,j (n) (i, j) P (n) p 0,0 (n) p 0,1 (n) p 0,2 (n) p 1,0 (n) p 1,1 (n) p 1,2 (n) = P (n) (5.12) p 2,0 (n) p 2,1 (n) p 2,2 (n)...... P (n) n transition probability matrix P (0) = I (5.13) 1 1, 2, 3,...,n 1
74 5 I n =1 (1) P 5.2 1 α α 0 0 P = 0 0 1 0 0 0 0 1 0 β 1 β 0 n P (n) i j 0 p i,j (n) 1 (5.14) n i p i,j (n) = 1 (5.15) j P (n) 1 n j π j (n) = i π i (0)p i,j (n) (5.16) π(n) =π(0)p (n) (5.17) P (n) (i, j) p i,j (n) (5.10) p i,j (n) =P (X(n) =j X(0) = i) = P (X(0) = i, X(n) =j) P (X(0) = i) (5.18) (5.6) P (X(0) = i) i 1,i 2,...,i n 1 P (X(0) = i 0,X(1) = i 1,...,X(n) =i n ) P (X(0) = i i 2 i 0 ) n 1 i 1
5.1 75 = p i0,i 1 p i1,i 2 p in 1,i n i 1 i 2 i n 1 (5.19) p i,j (n) = p i,i1 p i1,i 2 p in 1,j i 1 i 2 i n 1 (5.20) P (n) =P n (5.21) (5.17) (5.21) π(n) =π(0)p n (5.22) π(0) P n π(n) P n (5.21) i m k n j i k j m n E p i,j (m + n) = k p i,k (m)p k,j (n) (5.23) P (m + n) =P (m)p (n) (5.24) (5.23) (5.24) Chapman Kolmogorov equation 1 (5.24) n =1 P (m +1)=P (m)p (5.25) Kolmogorov s forward equation 1 m =1 P (n +1)=PP(n) (5.26) Kolmogorov s backward equation (5.25) (5.26) (5.21)
76 5 5.1 5.2 α =0.7 β =0.5 π(0) = [ 0.15, 0.05, 0.5, 0.3 ] 5.3 5.3 5.2 5.1 5.1.3 i j i j reachable i j i j j i i j i j i i i j j i i j j k i k equivalence class irreducible
211 e x 18, 26 FCFS 64 HOL 64 k 37 LCFS 64 M/G/1 150 M/M/1 137 M/M/1/K 143 M/M/S 124 M/M/S/K 130 M/M/S/S 117 M/M/S/S/N 110 n 73 n 73 PASTA 100 PDU 44 PR 64 PS 64 RR 64 t 84 t 84 2, 48 B 120 B 121, 191 C 127 C 128, 192 117 120 124 127 20 21 46 119 79, 91 110 114 112 9 43 19 6 39 13 94 6 81 13 11 27 152 152 77, 91 20 19 170 16 61 4, 65 56 22 76, 91 60 79, 93 10 47 48 48 40 30 30 31 23 63 1, 45 43 42 46 117 47 1, 48 114, 120 75, 86 75, 85 77, 91 160 60 61 47, 61 51, 61 140 113, 120 81, 93
212 5 81 5 58 21 61 61 101 78, 91 78 31 30 31 51 104 95 101 95 110 9 7 32 32 32 33 40, 65 40 71 89 40 72 73 77 46 23 50, 65 19 50 77, 91 6 44 45 40 71, 84 73, 84 88 88 7 201 47 47 82, 92 61 123 72 47 36 181 180 43 46 75, 85 24 179 81 56 71, 83 81, 91 181 46 76, 91 53, 60 56, 60 2, 50, 60 76, 91 8, 32 56 1 3 45 10 15 94 6 44 45 48 48 11 16 45 4, 44 78 19 5 5 165 52 122, 127 20 1 174 16 64 44 44 24 12 12 22 8 45 165 15 98 54, 98 17 46 12 12 159 46
213 47, 57 60 60 60 3, 60 61 2, 127 70 71, 83 63 70 13 59 85 44 44 23 61 5 47 64 188 26, 186 12 40 71 66 49, 65 77, 91 12 40 83 5 165 166 166
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