S Queueing Theory. M/G/1-PS queue. Samuli Aalto TKK Helsinki University of Technology. Lect_MG1-PS.ppt S Queueing Theory Fall 2009
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1 S Queueing Theory M/G/1-PS queue Samuli Aalto TKK Helsinki University of Technology Lect_MG1-PS.ppt S Queueing Theory Fall
2 Contents Preliminaries Queue length distribution for M/M/1-PS Queue length distribution for M/E K /1-PS Queue length distribution for M/G/1-PS Unfinished work distribution for M/E K /1-PS Unfinished work distribution for M/G/1-PS 2
3 M/G/1-PS queue Customers arrive according to a Poisson process at rate λ IID inter-arrival times exponential inter-arrival time distribution with mean 1/λ Customers are served by 1 server according to the PS service discipline IID service times a general service time distribution with mean 1/µ There are customer places in the system λ µ 3
4 Service discipline PS Processor Sharing (PS) customers are served simultaneously the service capacity is shared evenly among all customers ( fair queue ) preemptive, work-conserving, and non-anticipating discipline also known as Round Robin (RR) 4
5 Preliminaries (1) 5
6 Preliminaries (2) 6
7 Contents Preliminaries Queue length distribution for M/M/1-PS Queue length distribution for M/E K /1-PS Queue length distribution for M/G/1-PS Unfinished work distribution for M/E K /1-PS Unfinished work distribution for M/G/1-PS 7
8 M/M/1-PS queue Customers arrive according to a Poisson process at rate λ IID inter-arrival times exponential inter-arrival time distribution with mean 1/λ Customers are served by 1 server according to the PS service discipline IID service times exponential service time distribution with mean 1/µ There are customer places in the system λ µ 8
9 Exponential service time distribution S Exp ( µ ), µ > 0 P{S (x,x+h] S > x} = µh + o(h), where o(h)/h 0 as h 0 Value space: (0, ) µ PDF and CDF: f ( x) = µ e µ x F( x) = 1 e µ x 9
10 Moments and the Laplace transform S Exp ( µ ), µ > 0 Mean value: E[S] = 0 µx e -µx dx = 1/µ Second moment: E[S 2 ] = 0 µx 2 e -µx dx = 2/µ 2 Variance: D 2 [S] = E[S 2 ] E[S] 2 = 1/µ 2 Standard deviation: D[S] = D 2 [S] = 1/µ Coefficient of variation: C[S] = D[S]/E[S] = 1 Laplace transform: E[e -ss ] = 0 µ e -(µ+s)x dx = µ/(µ+s) 10
11 State transition diagram of X(t) Let X(t) denote the number of customers in the system at time t Assume that X(t) = i at some time t, and consider what happens during a short time interval (t, t+h]: with prob. λh + o(h), a new customer arrives (state transition i i+1) if i > 0, then, with prob. i(µ/i)h + o(h) = µh + o(h), a customer leaves the system (state transition i i 1) Process X(t) is clearly a Markov process with state transition diagram 0 λ µ λ 1 µ 2 Note that process X(t) is an irreducible birth-death process with an infinite state space S = {0,1,2,...} λ µ 11
12 12 M/G/1-PS queue Equilibrium distribution Detailed balance equations (DBE): Normalizing condition (N): 0,1,2,K, (DBE) = = = = = + + i i i i i i i i π ρ π ρπ π π µ π λ π µ λ (N) = = = = i i i i ρ π π ( ) 1 if, < = = = = ρ ρ ρ π ρ i i
13 Queue length distribution for M/M/1-PS (1) 13
14 Queue length distribution for M/M/1-PS (2) 14
15 Queue length distribution for M/M/1-PS (3) 15
16 Queue length distribution for M/M/1-PS (4) 16
17 Contents Preliminaries Queue length distribution for M/M/1-PS Queue length distribution for M/E K /1-PS Queue length distribution for M/G/1-PS Unfinished work distribution for M/E K /1-PS Unfinished work distribution for M/G/1-PS 17
18 M/E K /1-PS queue Customers arrive according to a Poisson process at rate λ IID inter-arrival times exponential inter-arrival time distribution with mean 1/λ Customers are served by 1 server according to the PS service discipline IID service times Erl(K, Kµ) service time distribution with K phases and mean 1/µ There are customer places in the system λ µ 18
19 Erlang service time distribution S Erl ( K, Kµ ), µ > 0 IID exponential phases in a series; S = S S K where S j Exp(Kµ) K = total number of phases µ = intensity of any single phase Value space: (0, ) PDF and CDF: f ( x) = F( x) Kµ = 1 K 1 ( Kµ x) ( K 1)! e K ( K x K µ ) ( K k)! k= 1 Kµ x k Kµ Kµ Kµ 1 2 K e Kµ x 19
20 Moments and the Laplace transform S Erl ( K, Kµ ), µ > 0 Mean value: E[S] = E[S 1 ] + + E[S K ] = K/(Kµ) = 1/µ Variance: D 2 [S] = D 2 [S 1 ] + + D 2 [S K ] = K/(Kµ) 2 = 1/(Kµ 2 ) Second moment: E[S 2 ] = E[S] 2 + D 2 [S] = (1+1/K)/µ 2 Standard deviation: D[S] = D 2 [S] = 1/(( K)µ) Coefficient of variation: C[S] = D[S]/E[S] = 1/( K) 1 Laplace transform : E[e -ss ] = E[e -ss 1 ] E[e -ss K ] = (Kµ/(Kµ+s)) K Kµ Kµ Kµ 1 2 K 20
21 Queue length distribution for M/E K /1-PS (1) Kµ Kµ Kµ 1 2 K 21
22 Queue length distribution for M/E K /1-PS (2) n 1 n 2 n K 1 2 K 22
23 Queue length distribution for M/E K /1-PS (3) n 1 n 2 n K 1 2 K 23
24 Queue length distribution for M/E K /1-PS (4) 24
25 Queue length distribution for M/E K /1-PS (5) 25
26 Queue length distribution for M/E K /1-PS (6) 26
27 Queue length distribution for M/E K /1-PS (7) 27
28 Queue length distribution for M/E K /1-PS (8) n 1 n 2 n K 1 2 K 28
29 Queue length distribution for M/E K /1-PS (9) n 1 n 2 n K 1 2 K 29
30 Queue length distribution for M/E K /1-PS (10) n 1 n 2 n K 1 2 K 30
31 Queue length distribution for M/E K /1-PS (11) 31
32 Queue length distribution for M/E K /1-PS (12) 32
33 Queue length distribution for M/E K /1-PS (13) 33
34 Contents Preliminaries Queue length distribution for M/M/1-PS Queue length distribution for M/E K /1-PS Queue length distribution for M/G/1-PS Unfinished work distribution for M/E K /1-PS Unfinished work distribution for M/G/1-PS 34
35 Queue length distribution for M/G/1-PS (1) 35
36 Queue length distribution for M/G/1-PS (2) 36
37 Queue length distribution for M/G/1-PS (3) 37
38 Queue length distribution for M/G/1-PS (4) 38
39 Queue length distribution for M/G/1-PS (5) 39
40 Queue length distribution for M/G/1-PS (6) 40
41 Queue length distribution for M/G/1-PS (7) 41
42 Queue length distribution for M/G/1-PS (8) 42
43 Queue length distribution for M/G/1-PS (9) 43
44 Contents Preliminaries Queue length distribution for M/M/1-PS Queue length distribution for M/E K /1-PS Queue length distribution for M/G/1-PS Unfinished work distribution for M/E K /1-PS Unfinished work distribution for M/G/1-PS 44
45 Unfinished work distribution for M/E K /1-PS (1) Kµ Kµ Kµ 1 2 K 45
46 Unfinished work distribution for M/E K /1-PS (2) 46
47 Unfinished work distribution for M/E K /1-PS (3) 47
48 Unfinished work distribution for M/E K /1-PS (4) 48
49 Unfinished work distribution for M/E K /1-PS (5) 49
50 Unfinished work distribution for M/E K /1-PS (6) 50
51 Unfinished work distribution for M/E K /1-PS (7) 51
52 Unfinished work distribution for M/E K /1-PS (8) 52
53 Unfinished work distribution for M/E K /1-PS (9) 53
54 Unfinished work distribution for M/E K /1-PS (10) 54
55 Unfinished work distribution for M/E K /1-PS (11) 55
56 Unfinished work distribution for M/E K /1-PS (12) 56
57 Unfinished work distribution for M/E K /1-PS (13) 57
58 Unfinished work distribution for M/E K /1-PS (14) 58
59 Unfinished work distribution for M/E K /1-PS (15) 59
60 Contents Preliminaries Queue length distribution for M/M/1-PS Queue length distribution for M/E K /1-PS Queue length distribution for M/G/1-PS Unfinished work distribution for M/E K /1-PS Unfinished work distribution for M/G/1-PS 60
61 Unfinished work distribution for M/G/1-PS (1) 61
62 Unfinished work distribution for M/G/1-PS (2) 62
63 Unfinished work distribution for M/G/1-PS (3) 63
64 Unfinished work distribution for M/G/1-PS (4) 64
65 Unfinished work distribution for M/G/1-PS (5) 65
66 Unfinished work distribution for M/G/1-PS (6) 66
67 Unfinished work distribution for M/G/1-PS (7) 67
68 References 68
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