EP2200 Queueing theory and teletraffic systems

Transcription

1 P00 Queueing theory and teletraffic ytem Lecture 9 M/G/ ytem Vitoria Fodor KTH S/LCN

2 The M/G/ queue Arrival proce memoryle (Poion( Service time general, identical, independent, f(x Single erver M/ r / and M/H r / are pecific cae, reult for M/G/ can be ued ule we can ue from the Marovian ytem x < for tability (ingle erver, no blocing Little: N T PASTA KTH S/LCN

3 The M/G/ queue ecall: M/M/: 0 μ μ At the arrival of the econd cutomer the time remaining from the ervice of the firt cutomer i till xp( M/G/: If we conider the ytem when a new cutomer arrive, then the remaining (reidual ervice time of the cutomer under ervice depend on the pat of the proce (on the elaped ervice time Conequently: the number of cutomer in the ytem doe not give a Marov chain KTH S/LCN 3

4 The M/G/ queue Solution method Average meaure N, T, etc. Mean value analyi Ditribution of the number of cutomer, waiting time, etc. Study the ytem at time point t 0, t, t,... when a cutomer depart, and extend for all point of time mbedded Marov chain KTH S/LCN 4

5 5 KTH S/LCN To calculate average meaure e tart with the average waiting time: the ervice of the waiting cutomer the remaining (or reidual ervice of the cutomer in the ervice unit Average remaining ervice time: Firt conditional average waiting time, then uncondition The M/G/ mean value analyi Pollacze-Khinchin mean formula ( ( (average include 0 remaining ervice time at tate 0 0, 0 (average waiting timefor cutomer arriving at tate,, 0,0,, 0 0,, q i i N r p p p p p

6 The M/G/ mean value analyi Pollacze-Khinchin mean formula e have to derive the average remaining ervice time : n: number of ervice in a large T number of Poion arrival: nt (ince the ytem i table T and n Note, have to include the 0 remaining ervice time at empty ytem. (t x x x x x 3 x 4 x 5 x n n n ( t i T n i ( i i Pollacze-Khinchin mean formula for the waiting time T t KTH S/LCN 6

7 7 KTH S/LCN From you can derive T, N, N q with Little theorem Comment: depend on the firt and the econd moment of the ervice time only Mean value increae with variance (cot of randomne The M/G/ mean value analyi Pollacze-Khinchin mean formula ( ( ( ( ( ( ( C x V V ( 0, /: / (, /: / C D M C M M x x

8 M/G/ waiting time ( ( C x Hyper-xp, (C x 4 rlang-4 (C x /4 M/M/ lambda KTH S/LCN 8

9 Group wor: The M/G/ mean value analyi Pollacze-Khinchin mean formula Conider the following ytem: Single erver, infinite buffer Poion arrival proce, 0. cutomer per minute Service proce: um of two exponential tep, with mean time minute and minute. Calculate the mean waiting time x x ( ( ( C x KTH S/LCN 9

10 Ditribution of number of cutomer in the ytem *** Comment: called a queue-length in the Virtamo note! *** The number of cutomer, N t i not a Marov proce the reidual ervice time i not memoryle e can model the ytem at departure time and extend the reult to all point of time: in the cae of Poion arrival the ditribution of N at departure time i the ame a at arbitrary point of time (PASTA if we are lucy then N t follow a dicrete time Marov proce at departure time ince thi dicrete time Marov chain i rather complex, we can expre the tranform form (z-tranform of the ditribution of the number of cutomer in the ytem. KTH S/LCN 0

11 Ditribution of number of cutomer in the ytem In the cae of Poion arrival the ditribution of N at departure time i the ame a at arbitrary point of time (PASTA PASTA i proved for arrival intant however, departure intant ee the ame queue length ditribution Let u follow N, N, N, that i, the number of cutomer in the ytem after departure N : number of cutomer after the departure of a cutomer V : number of arrival during the ervice time of cutomer, b(x i the ervice time ditribution, then: N α i N V N N V N 0 depend only on N and V, V i independent from i Dicrete time Marov Proce P( V i ( x i! e x b( x dx KTH S/LCN

12 M/G/ number of cutomer in the ytem α α α 3 i- i i i α 0 α i-j j xpreing the teady tate of the Marov-chain decribing N, we get the z- tranform of the ditribution of N Pollacze-Khinchin tranform form: Q ( ( ( ( z z B z * * B ( z z where: and B*( i the Laplace tranform of the ervice time ditribution. (S*( in the Virtamo note Ditribution of N with invere tranform, or moment of the ditribution..g., M/M/ KTH S/LCN

13 M/G/ ytem time ditribution ithout proof: Pollacze-Khinchin tranform form for the ytem time and waiting time: * ( ( * B ( T * * ( B ( ( B * ( where: x and B*( i the Laplace tranform of the ervice time ditribution. (S*( in the Virtamo note.g., M/M/ ytem time KTH S/LCN 3

14 Group wor again: Conider the ytem: The M/G/ mean value analyi Pollacze-Khinchin mean formula Single erver, infinite buffer Poion arrival proce, 0. cutomer per minute Service proce: um of two exponential tep, with mean time minute and minute. Give the Laplace tranform of the waiting time, calculate the mean waiting time * ( ( * B ( KTH S/LCN 4

15 M/G/ equirement for the exam: Derive and ue P-K mean value formula Ue P-K tranform form typically: for given ervice time ditribution give the tranform form, calculate moment Do not forget: M/M/, M/D/, M/ r / and M/H r / are pecific cae of M/G/ KTH S/LCN 5