S-38.3143 Queueing Theory M/G/1-LIFO-PR queue Samuli Aalto TKK Helsinki University of Technology Lect_MG1-LIFO-PR.ppt S-38.3143 Queueing Theory Fall 2009 1
Contents Preliminaries Queue length distribution for M/M/1-LIFO-PR Queue length distribution for M/E K /1-LIFO-PR Queue length distribution for M/G/1-LIFO-PR Unfinished work distribution for M/E K /1-LIFO-PR Unfinished work distribution for M/G/1-LIFO-PR Busy period distribution for M/G/1-LIFO-PR 2
M/G/1-LIFO-PR 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 LIFO-PR service discipline IID service times a general service time distribution with mean 1/µ There are customer places in the system λ µ 3
Service disciplines LIFO-PR Preemptive Last In First Out (LIFO-PR) customers are served one-by-one but not necessarily until completion when a new customer arrives, the current service is stopped and the new customer is taken into service when a server becomes free, the last arrived customer is taken into service thus, the customers in the system constitute a stack, and the system serves always the customer that has been waiting for the shortest time preemptive, work-conserving, and non-anticipating discipline also known as preemptive Last Come First Served (LCFS-PR) 4
Preliminaries (1) 5
Preliminaries (2) 6
Preliminaries (3) 7
Contents Preliminaries Queue length distribution for M/M/1-LIFO-PR Queue length distribution for M/E K /1-LIFO-PR Queue length distribution for M/G/1-LIFO-PR Unfinished work distribution for M/E K /1-LIFO-PR Unfinished work distribution for M/G/1-LIFO-PR Busy period distribution for M/G/1-LIFO-PR 8
M/M/1-LIFO-PR 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 LIFO-PR service discipline IID service times exponential service time distribution with mean 1/µ There are customer places in the system λ µ 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 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) 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. µh + 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 12
13 M/G/1-LIFO-PR queue Equilibrium distribution Detailed balance equations (DBE): Normalizing condition (N): 0,1,2,K, (DBE) 0 1 1 = = = = = + + i i i i i i i i π ρ π ρπ π π µ π λ π µ λ (N) 1 0 0 0 = = = = i i i i ρ π π ( ) 1 if, 1 1 1 1 1 0 0 < = = = = ρ ρ ρ π ρ i i
Queue length distribution for M/M/1-LIFO-PR (1) 14
Queue length distribution for M/M/1-LIFO-PR (2) 15
Queue length distribution for M/M/1-LIFO-PR (3) 16
Queue length distribution for M/M/1-LIFO-PR (4) 17
Contents Preliminaries Queue length distribution for M/M/1-LIFO-PR Queue length distribution for M/E K /1-LIFO-PR Queue length distribution for M/G/1-LIFO-PR Unfinished work distribution for M/E K /1-LIFO-PR Unfinished work distribution for M/G/1-LIFO-PR Busy period distribution for M/G/1-LIFO-PR 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 19
Erlang service time distribution S Erl ( K, Kµ ), µ > 0 IID exponential phases in a series; S = S 1 + + 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 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 21
Queue length distribution for M/E K /1-LIFO-PR (1) Kµ Kµ Kµ 1 2 K 22
Queue length distribution for M/E K /1-LIFO-PR (2) 1 2 K 23
Queue length distribution for M/E K /1-LIFO-PR (3) 1 2 K 24
Queue length distribution for M/E K /1-LIFO-PR (4) 1 2 K 25
Queue length distribution for M/E K /1-LIFO-PR (5) 26
Queue length distribution for M/E K /1-LIFO-PR (6) 27
Queue length distribution for M/E K /1-LIFO-PR (7) 28
Queue length distribution for M/E K /1-LIFO-PR (8) 29
Queue length distribution for M/E K /1-LIFO-PR (9) 1 2 K 30
Queue length distribution for M/E K /1-LIFO-PR (10) 1 2 K 31
Queue length distribution for M/E K /1-LIFO-PR (11) 1 2 K 32
Queue length distribution for M/E K /1-LIFO-PR (12) 33
Queue length distribution for M/E K /1-LIFO-PR (13) 34
Queue length distribution for M/E K /1-LIFO-PR (14) 35
Contents Preliminaries Queue length distribution for M/M/1-LIFO-PR Queue length distribution for M/E K /1-LIFO-PR Queue length distribution for M/G/1-LIFO-PR Unfinished work distribution for M/E K /1-LIFO-PR Unfinished work distribution for M/G/1-LIFO-PR Busy period distribution for M/G/1-LIFO-PR 36
Queue length distribution for M/G/1-LIFO-PR (1) 37
Queue length distribution for M/G/1-LIFO-PR (2) 38
Queue length distribution for M/G/1-LIFO-PR (3) 39
Queue length distribution for M/G/1-LIFO-PR (4) 40
Queue length distribution for M/G/1-LIFO-PR (5) 41
Queue length distribution for M/G/1-LIFO-PR (6) 42
Queue length distribution for M/G/1-LIFO-PR (7) 43
Queue length distribution for M/G/1-LIFO-PR (8) 44
Queue length distribution for M/G/1-LIFO-PR (9) 45
Contents Preliminaries Queue length distribution for M/M/1-LIFO-PR Queue length distribution for M/E K /1-LIFO-PR Queue length distribution for M/G/1-LIFO-PR Unfinished work distribution for M/E K /1-LIFO-PR Unfinished work distribution for M/G/1-LIFO-PR Busy period distribution for M/G/1-LIFO-PR 46
Unfinished work distribution for M/E K /1-LIFO-PR (1) Kµ Kµ Kµ 1 2 K 47
Unfinished work distribution for M/E K /1-LIFO-PR (2) 48
Unfinished work distribution for M/E K /1-LIFO-PR (3) 49
Unfinished work distribution for M/E K /1-LIFO-PR (4) 50
Unfinished work distribution for M/E K /1-LIFO-PR (5) 51
Unfinished work distribution for M/E K /1-LIFO-PR (6) 52
Unfinished work distribution for M/E K /1-LIFO-PR (7) 53
Unfinished work distribution for M/E K /1-LIFO-PR (8) 54
Unfinished work distribution for M/E K /1-LIFO-PR (9) 55
Unfinished work distribution for M/E K /1-LIFO-PR (10) 56
Unfinished work distribution for M/E K /1-LIFO-PR (11) 57
Unfinished work distribution for M/E K /1-LIFO-PR (12) 58
Unfinished work distribution for M/E K /1-LIFO-PR (13) 59
Unfinished work distribution for M/E K /1-LIFO-PR (14) 60
Unfinished work distribution for M/E K /1-LIFO-PR (15) 61
Contents Preliminaries Queue length distribution for M/M/1-LIFO-PR Queue length distribution for M/E K /1-LIFO-PR Queue length distribution for M/G/1-LIFO-PR Unfinished work distribution for M/E K /1-LIFO-PR Unfinished work distribution for M/G/1-LIFO-PR Busy period distribution for M/G/1-LIFO-PR 62
Unfinished work distribution for M/G/1-LIFO-PR (1) 63
Unfinished work distribution for M/G/1-LIFO-PR (2) 64
Unfinished work distribution for M/G/1-LIFO-PR (3) 65
Unfinished work distribution for M/G/1-LIFO-PR (4) 66
Unfinished work distribution for M/G/1-LIFO-PR (5) 67
Unfinished work distribution for M/G/1-LIFO-PR (6) 68
Unfinished work distribution for M/G/1-LIFO-PR (7) 69
Contents Preliminaries Queue length distribution for M/M/1-LIFO-PR Queue length distribution for M/E K /1-LIFO-PR Queue length distribution for M/G/1-LIFO-PR Unfinished work distribution for M/E K /1-LIFO-PR Unfinished work distribution for M/G/1-LIFO-PR Busy period distribution for M/G/1-LIFO-PR 70
Busy period distribution for M/G/1-LIFO-PR (1) 71
Busy period distribution for M/G/1-LIFO-PR (2) 72
Busy period distribution for M/G/1-LIFO-PR (3) 73
Busy period distribution for M/G/1-LIFO-PR (4) 74
Busy period distribution for M/G/1-LIFO-PR (5) 75
Busy period distribution for M/G/1-LIFO-PR (6) 76
Busy period distribution for M/G/1-LIFO-PR (7) 77
Busy period distribution for M/G/1-LIFO-PR (8) 78
Busy period distribution for M/G/1-LIFO-PR (9) 79
Busy period distribution for M/G/1-LIFO-PR (10) 80
Busy period distribution for M/G/1-LIFO-PR (11) 81
Busy period distribution for M/G/1-LIFO-PR (12) 82
Busy period distribution for M/G/1-LIFO-PR (13) 83
Busy period distribution for M/G/1-LIFO-PR (14) 84
Busy period distribution for M/G/1-LIFO-PR (15) 85
Busy period distribution for M/G/1-LIFO-PR (16) 86
References 87