Departure Process from a M/M/m/ Queue

Transcription

1 Dearture rocess fro a M/M// Queue Q - (-) Q Q3 Q4 (-) Knowledge of the nature of the dearture rocess fro a queue would be useful as we can then use t to analyze sle cases of queueng networs as shown. The ey result here s that the dearture rocess fro a M/M// queue s also osson wth the sae rate as the arrval rate enterng the queue. It should also be noted that the result of randoly slttng or cobnng ndeendent osson rocesses also yelds a osson rocess Coyrght Sanay K. Bose The result on the dearture rocess of a M/M// queue follows fro Bure s Theore. Ths theore states that - [A] The dearture rocess fro a M/M// queue s osson n nature. [B] For a M/M// queue at each te t the nuber of custoers n the syste s ndeendent of the sequence of dearture tes ror to t. [C] For a M/M// FCFS queue gven a custoer dearture at te t the arrval te of ths custoer s ndeendent of the dearture rocess ror to t. Coyrght Sanay K. Bose

2 Coyrght Sanay K. Bose 3 Te Reversblty roerty of Irreducble Aerodc Marov Chans Consder a dscrete te rreducble aerodc Marov Chan... n- n n... for whch the transton robabltes are gven to be. Now consder the sae chan bacwards n te.e. the chan... n n Ths would also be a Marov Chan snce we can show that * State Transton robablty of the Reverse Chan Coyrght Sanay K. Bose 4 The Marov Chan s consdered to be te reversble for the secal case where *. The reverse chan wll have the followng roertes - The reversed chan s also rreducble and aerodc le the forward chan The reversed chan has the sae statonary state dstrbuton as the forward chan The chan s te reversble only f the detaled balance equaton holds for

3 How can we handle queues where the servce te dstrbuton s not exonental? [A] If we can exress the actual servce te as cobnatons of exonentally dstrbuted te ntervals then the Method of Stages ay be used. (Secton.9) [B] The M/G/ queue and ts varatons ay be analyzed. (Chaters 3 and 4) [C] Aroxaton ethods ay be used f the ean and varance of the servce te are gven. (GI/G/ aroxaton of Secton 6.) Coyrght Sanay K. Bose 5 Method of Stages Stage /µ Stage /µ Consder a M/-// exale where the actual servce te s the su of two rando varables each of whch s exonentally dstrbuted. State of the syste reresented as (n ) where n s the total nuber of custoers n the syste where the custoer currently beng served s at Stage n... State () reresents the state when the syste s ety () ( ) ( ) µ µ State Transton µ µ Dagra of the Syste ( ) ( ) Coyrght Sanay K. Bose 6 3

4 Balance Equatons for the Syste µ ( µ ) ( µ ( µ ) ( µ etc... ) ) µ µ µ µ 3 (.38) These Balance Equatons ay be solved along wth the arorate Noralzaton Condton to obtan the state robabltes of the syste. Once these are nown erforance araeters of the queue ay be arorately evaluated. Coyrght Sanay K. Bose 7 The ethod llustrated for the M/-// exale ay be extended for the followng tyes systes.. Have stages of servce tes - ore rows n the state transton dagra. Fnte Nuber of Watng ostons n the Queue - ae the arrval rate a functon of the nuber n the syste and ae t go to zero once all the watng ostons have been flled 3. Multle Servers - aroxate ths by allowng ore than one ob to enter servce at a te 4. More General Servce Te Dstrbutons - see next slde Coyrght Sanay K. Bose 8 4

5 For ore general servce te dstrbutons the Method of Stages ay be used f the Lalace Transfor of the df of the servce te ay be reresented as a ratonal functon of s L B (s)n(s)/d(s) wth sle roots. Entry Stage µ α α -α -α Stage µ Ths leads to - L ( s) ( α B Ext Wth ultle stages le ths the L.T. of the servce te df wll be of the for - ) α... α ( α ) µ s µ L ( s) B β β s µ Coyrght Sanay K. Bose 9 Gven a servce te df as L B (s)n(s)/d(s) wth sle roots -. Obtan the ultle stage reresentaton n the for shown earler. Draw the corresondng state transton dagra and dentfy the flows between the varous states 3. Wrte and solve the flow balance equatons along wth the noralzaton condton to obtan the state robabltes 4. Use the state robabltes to obtan the requred erfroance araeters Coyrght Sanay K. Bose 5

6 Queues wth Bul (or Batch) Arrvals (Secton.) M [] osson Batch Arrval rocess Batches arrvng as a osson rocess wth exonentally dstrbuted nter-arrval tes between batches Batch sze Nuber of obs n a batch (rando varable) Average Batch Arrval Rate β r r obs n a batch r. β ( β r z r β r rβ r r Coyrght Sanay K. Bose The M [] /M/ Queue µ ( for ) µ β for Balance Equatons Though these ay be solved n the standard fashon we wll consder a soluton aroach for drectly obtanng ( the Generatng Functon for the nuber n the syste. For ths we would need to ultly the th equaton above by z and su fro to. ( µ ) z µ z z β z ( n n n z Coyrght Sanay K. Bose 6

7 Slfyng we get µ ( µ )[ ( ] [ ( z µ ( ( µ ( z[ β( ] z] ( β ( β Defne ρ as the offered traffc µ Note that () s effectvely the sae as the Noralzaton Condton. Usng ths we get ρ µ ( ρ)( ( µ ( z[ β ( ] Therefore (.4) We can nvert ( or exand t as a ower seres n z to get the state robablty dstrbuton. The ean nuber N n the syste ay be drectly calculated fro ( as - d( ρ( β β ) N (.43) dz ( ) z ρ Coyrght Sanay K. Bose 3 The M [] /-/-/K Queue Batch Arrval Queue wth Fnte Caacty For oeratng queues of ths tye one ust also secfy the batch accetance strategy to be followed f a batch of sze or ore arrrves n a syste where the nuber of watng ostons avalable s less than. artal Batch Accetance Strategy (BAS) Whole Batch Accetance Strategy (WBAS) Randoly choose as any obs fro the batch as ay be accoodated n the buffer Accet the batch only f all ts obs ay be accoodated; otherwse reect all obs of the batch Coyrght Sanay K. Bose 4 7

8 M [ /M/-/- tyes of queues ay be oerated and analyzed under ether the BAS or the WBAS strategy See Secton. where ths analyss s done for a M [ /M/s/s queue. The state dstrbuton for ths queue are gven by φ... s µ (.46) where φ β... Coyrght Sanay K. Bose 5 8