Time Dependent Queuing

Transcription

1 Time Depede Queuig Mark S. Daski Deparme of IE/MS, Norhweser Uiversiy Evaso, IL 628 Sprig, 26

2 Oulie Will look a M/M/s sysem Numerically iegraio of Chapma- Kolmogorov equaios Iroducio o Time Depede Queue Aalyzer

3 Chapma Kolmogorov Equaios M/M/s queue 2 s- s s s2 3 2 (s-) s s s s Sae Trasiio Diagram () () () () () () () () () () () () [ ] () () () () d d d d s,...

4 Firs-order differece equaios () () () () () ( ) ( ) () () () () ( ) () () () () () { } () () () () () () () [ ] () ( ) ( ) () () () () () () [ ] () ( ) () () () () () () () [ ] () { },...,...,... Δ Δ Δ Δ Δ Δ Δ Δ d d d d You ca ge firs order differece equaios as show o he ex slide

5 Firs-order differece equaios ( ) () () () () () { } ( ) () () () () () () () [ ] () { },... Δ Δ Δ Δ Give some iiial esimae of he sae probabiliies a ime, we ca use hese equaios o esimae he sae probabiliies a some ime Δ ad so o.

6 racical implemeaio Make sae space fiie (max saen) Adjus equaio for N accordigly Divide he day io small ime slices E.g., use Δ6 secods or less. Begi wih seady sae esimae of probabiliies Icreme s as eeded o ge seady sae durig ay ime slice ha has s<.

7 racical implemeaio Use fourh-order Ruge-Kua o sep bewee ime slices ad o firs-order Euler as show above If ay () becomes egaive, se i o Reormalize all sae probabiliies a each ime period Compue larges % chage i probabiliies (for prob>. for example)

8 racical implemeaio If ay sae probabiliies go egaive, sar process over wih smaller Δ If larges % chage oo big, cycle hrough probabiliies agai. Repea as eeded.

9 Which sae probabiliies ifluece (Δ) -, Geeral case,,δ,,, N-, N,,Δ For sae For sae N N,Δ

10 Euler vs 4 h Order Ruge-Kua rob Euler rob Ruge-Kua Slope based o esimae Slope based o 4 esimaes, givig accuracy of 4 h order expasio Δ Δ

11 Euler vs 4 h Order Ruge-Kua ( ) ( ) ( ) [ ] ( ) () () () () () [ ] () () [ ] () () () [ ] () () () [ ] () () () [ ] 2, 2 2.5, 2.5, 2, 3 2 2, Ruge- Kua Euler k f k k f k k f k f k k k k k f Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ () () ( ) ( ) ( ) () () () () () () () [ ] () () () () () () N f f f N N N N N,...

12 Graphical represeaio of soluio procedure 2 5 Approach Evolve MFs over ime Demad 5 Cycle back as eeded Time

13 Time Depede Queue Aalyzer Be sure your copy looks like his. Earlier versios had a error i he code. You should see Class Versio udereah my address.

14 Major blocks of ipus/oupus Be sure your copy has he versio umber i he op lie. Earlier versios of he code coai a error. Demad ifo Service ifo Cos ipus Compuaioal ifo rimary graph oupu Sae prob graphs Summary oupu Corol Buos

15 Se imes form Duy imes Graph of duy imes

16 Sar imes page Se sarig imes for full ime ad parime employees Graph of demad ad service capaciy Click o reur o mai meu

17 Seady sae base case oupu Service capaciy Demad Mea # i Sysem Noe ha peak umber i he sysem correspods i ime o peak demad

18 Time Depede Case Time Dep. # i sysem Noe ha peak umber i he sysem uder ime depede codiios is shifed o he righ of he peak demad ad is lower ha he seady-sae peak. Will look a MF a 4

19 Compariso of MFs a 4 Time Depede Seady-sae Noe ha ime depede probabiliies are higher a low ed reflecig he smaller mea a 4.

20 (wai) Noe ha peak (wai) i he sysem uder ime depede codiios is shifed o he righ of he peak demad ad is lower ha he seady-sae peak.

21 More complex behavior Noe ha he program does o require seady sae codiios i each ime slice.

22 Summary Time depede queuig aalysis is impora eaks are shifed o he righ of seady sae peaks Does o require seady-sae codiios i all periods.