Queueing Networks II Network Performance

Transcription

1 Queueng Networks II Network Performance Davd Tpper Assocate Professor Graduate Telecommuncatons and Networkng Program Unversty of Pttsburgh Sldes 6 Networks of Queues Many communcaton systems must be modeled as a set of nterconnected queues whch s termed a queueng network. Systems modeled by queueng networks can roughly be grouped nto four categores Open networks losed networks Networks wth populaton constrants (also called Loss Networks) Mxed network 1

2 Networks wth Populaton onstrants (Loss Networks) onsder M queue system ustomers arrve from outsde the network accordng to a Posson process wth rate to queue. Exponental servce dstrbuton wth rate at queue Total system sze (watng space ) s B Smple example: M output queues at an output buffer of a packet swtch. 1 1 m m 3 Networks wth Populaton onstrants Ths process s a fnte state space M dmensonal Markov process wth state space m S ( n1, n,, nm ): 0 n B ; n B 1 The steady state probablty P( n) lm Pn~ t n n~ t n n~ 1( ) 1, ( ),, m( t) nm t P(n) has a product form m 1 n P( n) G 1 where G s the normalzaton constant found by P( n) 1 m ns G n ns 1 4

3 Networks wth Populaton onstrants From P(n) one can determne varous mean performance measures. L Average number of customers n queue. L B j P( n) j0 n j; ns LN M L 1 W Average delay at queue found by Lttle s Law 5 Networks wth Populaton onstrants Example: B = 3, M = State dagram (n 1, n ) : S = { (n 1, n ) ; 0 n 1 3, 0 n 3, n 1 +n 3 } 1 m 1 1 n1 n P( n 1, n) n G G Let 1 = 0.5, = 1, = 1, = = 0.5, = G = G

4 Networks wth Populaton onstrants 1 n1 n P( n 1, n) (0.5) (1) 6.15 P(, 1) = L1 n 1 P ( j, n ) 1 P n1 0 j n1 ; ns L 1 = Smlarly L = LN = P (1,0) P (1,1) P (1,) 1P (,0) P (,1) 3 (3,0) 7 Networks wth Populaton onstrants Multrate loss system : Mult-dmensonal loss systems onsder a sngle lnk n a mult-rate crcut swtched network Varous servces are offered and each servce has dfferent characterstcs (call arrval rate, holdng tme, bandwdth.) Assume K types of connectons each type arrves accordng to a Posson process rate and have holdng tme exponentally holdng tme wth a rate (results hold for general holdng tme.) Each type connecton requres m basc unts of bandwdth. The total bandwdth avalable s unts. hapter 7 or ITU Teletraffc Handbook 8 4

5 Networks wth Populaton onstrants Let n ~ ( t ) = number of type connecton n system at tme t. K dmensonal Markov process wth fnte state space S 0 n / m and nm K 1 n~ ), ~ ( ),, ~ 1 ( t n t nm( t) 9 Loss Networks = 10, K =, m 1 = 1, m = n n TELOM 10: Network Performance

6 Networks wth Populaton onstrants The steady state probabltes P( n n nk Pn~ t n n~ t n n~ 1,,, ) lm 1( ) 1, ( ),, K ( t) nk t m The product form exsts where K 1 n when K=1, get Erlang B model M/G// P( n1, n,, nk ) G( k) n 1! Sometmes called Generalzed Erlang eq K n G( k) n! n S 1 onnecton blockng rates PB P( n1, n,, nk ) ns K n where type blocked sum over states where m n jm j j1 11 Networks wth Populaton onstrants PB P(0,5) P(1,4) P(,4) P(3,3) P(4,3) P(5,) P(6,) P(7,1) P(8,1) P(9 9,0) P (10,0) 0) PB1 P(0,5) P(,4) P(4,3) P(6,) P(8,1) P(10,0) Numercal example: = 48, K=, k=1 voce 64 Kbps m 1 = 1 k= H 3 vdeo 384 Kbps m = 6 1 = 15, = 0.15, 1 = 1, = 0.5 K m Offered load 30 1 PB 1 = 0.048, PB =

7 rcut Swtched Networks Model each lnk by Erlang B or mult-class loss system How to determne end-to-end blockng? onsder case of sngle traffc class (e.g., voce) of N nodes and L lnks. Let be the capacty of lnk and a be the load n Erlangs at lnk, B (, a ) s the call blockng rate on lnk End to End all Blockng along a path P j End to end Blockng 1 (1 B (, a )) P j Assumes load ndependent on each lnk f load of sngle flow s small fracton on each lnk O.K. approxmaton 13 End to End Blockng Two T1 lne example, offered load Erlangs, 1 = = 4 1 Destnaton B 1 = B(0, 4) =.066, B = B(6, 4) =.189 Estmate end to end blockng 1 (1- B 1 )(1- B ) =.45 Note assumes traffc s ndependent on each lnk an mprove approxmaton by reducng the load on lnk to account for blockng at lnk 1 Thus load on lnk = 0*(1- B 1 ) + 6 = 4.68 Erlangs B = B(4.68, 4) = Ths s called a Modfed Load Approxmaton or Reduced Load Approxmaton Yelds End to end blockng 1 (1- B 1 )(1- B ) =.164 Assumes load from source 1 thnned on frst lne before beng carred on second lne 14 7

8 Erlang Fxed Pont Approxmaton Let A be the offered load n Erlang sfrom source to a path from to j In realty the number of calls actve from source to destnaton must be the same on each lnk along the path as sgnalng wll reserve end to end resources before call s connected Use reduced load approxmaton to get estmate of load at each lnk l a A (1 B (, a )) /(1 B (, a )) s P j s s s Get a set of coupled non-lnear equatons that are solved teratvely for a soluton untl B converge at each lnk - ntalze by computng every lnk ndependently an be extended to mult-class of traffc, routng etc. See hapter 5. ``K. Ross, Multservce Loss Models for Broadband ommuncaton Networks, Sprnger-Verlag, Loss Networks Many generalzatons of Loss Systems General Servce Tmes, PS queueng dscplne etc. Several algorthms for effcent computaton of G(K) 16 8

9 Networks of Queues Systems modeled by queueng networks can roughly be grouped nto categores Open networks Networks wth populaton constrants losed networks Looked at cases where state probabltes P(n) have a product form soluton. Where determned from normalzaton condton m P( n) n 1 What about networks wthout product form? Lmted results manly specal cases or approxmatons 17 Remember G/G/1 KLB Approxmaton KLB approxmaton based on two moments of the arrval and servce tme dstrbutons. ( ) a s J L (1 ) where J scalng factor Ths approxmaton s often used to determne the effects of ncreased utlzatons on systems where measurement data s avalable to determne a and s (1 )(1 a ) 3( a s ) e J (1 )( a 1) ( 4 a s ) e ; a 1 ; a

10 QNA Assume arbtrary network of M queues, defne r ( m1) Total mean customer arrval rate to queue. Mean arrval rate from outsde of network to queue, external arrvals r j Routng probablty customer leavng queue goes to queue j. Probablty customer leavng queue exts the network. Mean servce rate at queue. o Squared coeffcent of varaton of outsde arrvals als to. s Squared coeffcent of varaton of servce process at. Squared coeffcent of varaton of arrval process at. A TELOM 10: Network Performance 19 Open Network of G/G/1 Queue Whtt s method for open s network of G/G/1 queues - Queueng Network Analyss (QNA) The basc dea s to use the KLB G/G/1 two moment approxmaton at each queue n the network. The model of queue s smlar to the arbtrary queue studed n Jackson networks. 1 r 1, o r (m+1) ( m1) r 1 r, A, s r m r m r r m 0 10

11 QNA As n the Open Jackson network case, fnd mean arrval rate at each queue by the flow conservaton equaton m r j j 1 ( I R) j1 To apply KLB equaton need A at each queue. Ths requres the applcaton of three approxmatons (smlar to Jackson network approach) for 1. Departure process approxmaton. Spttng process approxmaton 3. Mergng process approxmaton 1 QNA - Departure Process Approxmaton, a, s Mean departure rate = d = Squared coeffcent of varaton of departure process. Based on renewal process approxmaton d s ( 1 ) A 11

12 QNA Splttng Process Approxmaton If a process wth mean and s probablstcally splt nto K stream wth probabltes K p p 1 1, p 1 p 1, 1, We can approxmate p p (1 p ) as below pk k, k 3 QNA Mergng Process Approxmaton K The of a merger of K streams s approxmated by 1 1, 1, M, s, 1W W j1 j k, k W 1 4(1 ) M j ombnng the three approxmatons to determne A at each queue 4 1

13 QNA Mergng Process Approxmaton A M jr j 1 W W o rj j sj (1 j ) j1 Aj (1 r j ) yelds a system of lnear equatons to solve for A where 1 ( ) 1 4(1 ) 1 M r j W 1 j1 Once A approxmaton solved can treat each queue ndependently and determne the mean metrcs for each queue from the KLB approxmaton for G/G/1 queue and the network-wde measures LN, WN, etc 5 QNA Summary Gven, o,, S, r j Solve for 1 ( I R) Then solve for A A M j r j 1 W W o r j j sj (1 j ) j 1 1 ( ) 1 4(1 ) 1 M rj W j 1 Use KLB approxmaton to fnd mean behavor for each queue Aj (1 r QNA approxmaton tends to do pretty well on network-wde measures LN, WN, etc, but not so well for ndvdual queues. QNA mplemented n several software packages (QNAP), (RAQS) j ) 6 13

14 Example onsder tandem queueng model below. ustomers arrve to the frst queue accordng to a Posson process wth mean rate 1 =10 1.0, and o1 =1 Outsde customers arrve to the second queue accordng to a determnstc process wth mean rate = 1.0, and o = 0 Servce process at queue one s Erlang dstrbuted wth 1 = 1., S1 = ½ Servce process at queue two s exponental wth rate =., S = 1 1 ( I R ) 1 = 1.0, =, 1 = 1/1. =.833 = /. = Example From the fgure A1 = o1 = 1 1 M ( r ) 1 4 (1 ) j W 1 j 1 A M jr j 1W W o rj jsj (1 j ) j1 W = 1/.975 = Aj (1 r ) j A =.3093 From KLB equaton get ( a s ) J L (1 ) J 1 = 1, L 1 = , J =.903, L =.1774 where J scalng factor (1 )(1 a ) 3 ( ) e a s ; 1 J a Get LN = L 1 + L = , N = (1 )( a 1) ( 4 ) WN = LN/N = a s e ; a

15 Summary Networks wth Populaton onstrants Mult-class lnks Mult-rate lnks Networks of mult class or rate systems QNA Approxmaton for G/G/ 1 networks 9 15