Optimization of the CBSMAP Queueing Model

Transcription

1 July London UK Optiization of the CBSMAP Queueing Model Kondrashova EV Kashtanov VA Abstrat The present paper is devoted to the researh of ontrolled queueing odels at ontrol of CBSMAP-flow Controlled Bath Sei-Marov Arrival Proess (Kashtanov Kondrashova 22) The ontrol is based on the theory of ontrolled sei-arov proesses and used for the syste optiization The ontrol is arried out using the hoie of the net bath type Inde Ters ontrolled queuing odels optiization ontrolled proess In the further we shall use the following designations: P{ ξ θ t u B / ξ i} n n n n Q (2) ( t B) i For eah state i the set of ontrols U i and σ-algebra A i of subsets of this set U i is given At t and B=U i we obtain the transition probability T I INTRODUCTION he funtioning of different systes an be desribed using queueing odels Appliation of ontrol is used to inrease the effiieny of the syste funtioning In the present paper the proess of syste funtioning is investigated using ontrol by the arrival flow The Controlled Bath Sei-Marov Arrival Proess is a generalization of the BMAP-flow [9]-[] BMAP-flow is good for odeling of data-flows in teleouniation networs Define CBSMAP-flow After holding in the state oes to an end the Controlled Sei-Marov proess ups to the other state and the bath of queries of CBSBMAP-flow will be generated II CONTROLLED SEMI-MARKOV PROCESS Controlled Sei-Marov proess X ( t) { ξ ( t) u( t)} is defined using hoogeneous three-diensional arov hain (Kashtanov 2) n n n n n n ( ξ θ u ) n ξ E θ R [ ) u U whih is defined by transition probabilities of a speial type P{ ξ θ tu B/ ξ i θ τ u u} n n n n n n P{ ξ θ tu B/ ξ i} n n n n i E t τ R uu BA p P{ ξ i θ u U} i Manusript reeived Marh 2 23; revised April 4 23 KEV Author is with the Departent of Higher Math Mosow State Institute of Eletronis and Matheatis National Researh University Higher Shool of EonoisBolshoi Trehsvyatitelsi per 3 Mosow Russia evondrashova@hseru KVA Author is with the Departent of Higher Math Mosow State Institute of Eletronis and Matheatis National Researh University Higher Shool of EonoisBolshoi Trehsvyatitelsi per 3 Mosow Russia () p Q ( U ) P{ ξ / ξ i} (3) i i i n n for ebedded Marov hain Use definition Q ( t B) i Q ( B) li Q ( t B) i t i (4) P{ ξ n u B/ ξ n n i} Then G ( ) Q ( B ) P { B / i } (5) i i n n E and Q ( t B ) Q ( t u ) G ( du ) (6) i i i B Q ( t u) P{ ξ θ t / ξ iu u} (7) i n n n n Thus hoogeneous Controlled Sei-Marov proess an be set by faily of atries set of probability easures and initial distribution of probabilities p i ie tr + u U i BA i Faily of atries { Q ( t u )} is a Sei-Marov ernel i of ontrolled Sei-Marov proess and faily of probability easures G { G ( B) G ( B) G ( B)} is 2 N faily of ontrolling easures The ounting proess ν() t is defined in following way ν ( t) sup{ n : θ t} θ n The Controlled Sei-Marov proess is defined as Xt () {() ξ t u()} t ξ ξ ν ut () u ν() t

2 July London UK Proess ξ ( t ) oinides with a standard Sei-Marov proess The seond oponent of ontrolled Sei-Marov proess u ( t ) defines a traetory of aepted deisions It is possible to define one ore way to give ontrolled Sei-Marov proess It is neessary to set: Marov hoogeneous ontrol strategy G { G ( B) G ( B) G ( B)} 2 N harateristis of ontrolled Marov hain - initial distribution p P { ξ i } i E p and a atri i i ie of transition probabilities p ( u ) P { ξ / ξ iu u } ; i n n n onditional distributions of intervals F ( t u) P{ θ t / ξ ξ iu u} i n n n n III CBSMAP MODEL A Assuptions Consider that the final bath of ustoers of -th type arrives at the oent of CSMP (Controlled Sei-Marov Proess) transition in state in queue odel the nuber of queries in bath ν is defined by generating funtion M Φ ( z) z p ( ) (8) ( ) p ( ) is a probability ( the nuber of -th type ustoers in bath is ) M a aial nuber of ustoers in bath of -th type Forulate the iportant assuptions at whih the further researhes will be arried out Custoers of the sae type arrive in the syste (subsyste) eah of whih funtions irrespetive of other subsystes Designate as systehe subsyste whih is arrying out servie of queries of -th type Notie that the proess of servie in eah syste is realized irrespetive of other syste states however funtioning of the systes is oordinated with the general arrival flow 2 Between the net oents of the hange of the CSMP states ustoers do not arrive in the syste only the proess of servie in subsystes is arried out Proess of servie in -th subsyste is haraterized by nuber ν of the ustoers whih are being in the subsyste during the oent t Note that there are several types of adission disipline Three adission disiplines are nown: - partial adission when only a part of the bath orresponding to the nuber of free plaes in the buffer is allowed to oin the syste; - oplete adission when the whole bath is allowed to enter the syste if there is at least one free plae in the buffer; - oplete reetion when the whole bath is reeted B Desription of the odel Algorith Desribe the odel The given syste onsists of N subsystes The subsyste of -th type N in designations of Kendall s notation an be desribed as follows: CBSMAP/M /n /N - СBSMAP eans that the arrival flow is ontrolled flow defined earlier; - Sybol М eans that servie duration of ustoer in a subsyste is eponentionally distributed with paraeter μ ; -Sybols n and N define the quantity of the servie buffers and the nuber of plaes in the queue n and N aordingly In lassifiation of queueing systes the syste an be onsidered as a Controlled Sei-Marov syste as its evolution is defined with Controlled Sei-Marov Proess For onstrution of CSMP desribing the evolution of the syste it is neessary to arry the following algorith: Define Marov oents Define the states of Sei-Marov Proess; Define ontrol set and ontrol strategy; Define Sei-Marov ernel and a atri of transition probabilities of ebedded Marov hain; Construt inoe funtional on the traetories of CSMP; Define optiu strategy of ontrol In the given odel the Marov oents are the oents of arrivals of any type ustoers in syste In ase of -th type ustoers arrival the given ustoers are taen on servie to a subsyste of -th type and in other subsystes the bath of zero quantity "arrives" The syste states are defined using a vetor ( il l 2 l N ) i - a state of an arrival flow (at Marov oent the bath of i-th type ustoers arrives) l - quantity of queries in a subsyste of -th type M N n l i M N n l i i i i n and N - aordingly quantity of servie hannels and quantity of plaes in the queue in Syste () i E { 2 N} The quantity of ustoers in a subsyste is final and depends on adission disipline and struture of queueing odel Therefore l E { 2 M N n } and ( il l l ) E E E 2 N N Enter the following designations: ( l l l ) l 2 N ; E E E E N The transition fro state ( il l l ) to state 2 N ( l ' l ' l ' l ' ) with positive probability ours 2 N if l ' l that is true to all subsystes eept for

3 July London UK a subsyste of -th type there is only a servie ustoers whih an be presented as proess of death proess aording to the quantity of ustoers in these subsystes is not ore than the quantity of ustoers in subsystes at previous Marov oent of bath arrival Note that the syste ontrol is arried out using ontrol of arrival-flow at the oents of SMP states hange at the Marov oents Reind that ontrol Marov strategy G ( G ( u) ( i l ) E ) depending only on a urrent () il state of ontrolled proess is a set of the probability easures given for eah state ( il) E on σ - algebra of subsets of deisions set U () il C Control set As it was ared above the proble of ontrol with arrival flow is investigated The flow is desribed by Controlled Sei-Marov Proess Construt a ontrol set At Marov oents (transition oents state ( il) E ) the type of queries is hosen Then equality U () E { 2 N} is fair and the il probability easure on disrete set E { 2 N} is defined with a set of probabilities G ( ) P{ u( t) / ξ( t) ( i l )} p () il () il p p E { 2 N} () il () il E (9) Desribe the designations used in previous equality ( ) The probability p s( ) is defined as follows ( ) ( ) P{ ν ( t) s / ν ( ) } s s p s ; p s () probability of that in -th subsyste during tie t (between the Marov oents) (-s) queries are operated provided that during the initial oent there were queries in Syste If Syste () is a syste without queue N = ( ) s μt ( s) sμt () ( ) s p t C e e n s () n - the nuber of hannels μ - paraeter of eponential distribution of the servie duration For other obinations of paraeters n s required probabilities are equal to zero If Syste () is a single-hannel syste with queue n = N > then s ( μ t) μt s p e N s ( s)! the deision aepted at Marov oent t is designated ut (); The Sei-Marov ernel is a probability of that the Sei- Marov proess will pass in state ( l ) (at the Marov oent the bath of -th type arrives and there will be l queries in subsystes) and the tie of this transition will not surpass t provided that the proess stays in state ( i ) and in the state the deision uu ( i ) fro the ontrol set is aepted Designate the probability as Q ( t u ) ( i)( l ) In ase U ( ) E { 2 N} the type of queries i whih arrive to the syste is hosen: uu E { 2 N} ( i) E { 2 N} ( i)( l ) E t Q ( i)( l ) ( t u) t ( ) ( ) ( v) s N ns v lv i s v ( ) ( ) μ t ( μ t) p p e l l l l! N s l (2) For other obinations of paraeters n s required probabilities are equal to zero The general ase if Syste () is a ultihannel syste with queue n N then s μt ( s) sμt s nμt s C ( e ) e n s ( n μ t) t ( s)! e s n n s μ ( t ) ( n s) sμ ( ) n μ nμ n ( n )! C ( e ) e n μ e d n s p ( ) p ( l s) p ( ) df ( ) u ; ( ) Probabilities p () l - probability to aept queries u fro bath of -th type at presene of epty seats depending on an adission disipline It is defined using p

4 July London UK following equalities In ase of partial adission p ( s) P{in( ν ) s} p ( s) s 2 ; p () l s ; l s In ase of oplete adission p ( s) s ; p ( s) s In ase of oplete reetion (3) (4) ( l ) and the deision u is ade The onditional atheatial epetation of the saved up inoe R ( u ) depends on the inoes and the ( i)( l ) harges of syste wor Enter the onstants desribing these inoes and harges: - the inoe reeived for servie of one ustoer; 2 - a payent for a tie unit during the woring of one devie during servie; 3 - a payent for a tie unit of idle tie of one devie; 4 - a payent for a tie unit for staying in queue for one query; 5 - a payent for one lost query of -th type Then p ( s) s ; p ( s) p ( l) s ; (5) l s ( ) ( ) ( i)( l ) R ( u) C ( l u) 4 s2 s 5 5 C ( n l u) C ( l u) (6) F ( ) - probability of that the following bath of i ustoers will arrive in syste before the oent provided that it is a bath of -th type and the previous bath was a bath of i-th type The Sei-Marov ernel then ( ) ( ) ( i)( l ) ( i)( l ) ( i) U ( i ) Q Q ( t u) G ( du) t ( ) ( ) ( v) ( i) s Nns v lv i s v p p ( ) p ( l s) p ( ) df ( ) ( ) ( ) C ( l u) M( / () ( ) l u() u) C ( n l ) u 2 ( ) ( ) ( ) ( ) 2 M[in( n ν ()) t dt/ ν () ξ () l () u u] C ( n l ) u 3 () () () () 3 M[( n in( n ν ()) t dt/ ν () ξ () l () u u] At t a atri of transition probabilities of ebedded Marov hain ( ) ( ) li ( i)( l ) ( i)( l ) p Q t U ( i ) ( i)( l ) Q ( u) G ( du) ( i) ( ) ( ) ( v) ( i) s Nns v lv i s v p p ( ) p ( l s) p ( ) df ( ) D Inoe funtional Further it is neessary to define funtions R ( u ) - a onditional atheatial epetation of ( i)( l ) the saved up inoe in Syste () provided that proess stays in state ( i ) through tie t it will pass in state C ( n l ) u 4 () () () () 4 M[a( ν n ) dt/ ν () ξ () l () u u] C ( l u) 5 ( ) ( ) ( ) M( / () () ( ) l u() u) Notie that ξ ( ) ( ) - the nuber of ustoers at the Marov oent ν t - Marov proess of destrution on the period t ( ) ( l ) () Designate S - the atheatial epetation of the saved up inoe in subsyste during tie t

5 July London UK provided that the proess starts fro the state ( l ) [3] For the funtional S () ile () ile s S ( ) ( ) () il () il ( l ) ( ) ( ) () il () il ( ) ( ) ( i) ( i)( l ) ( l ) E [ Q ( t)] dt the following equality is fair (7) (8) atheatial epetation for the tie of ontinuous staying of the proess in state ( i ); ( i) ( ) ( ) ( i)( l ) ( i)( l ) ( i) [ R ( u) dq ( u)] G ( du) uu ( i) ( l ) o s (9) atheatial epetation of the saved up inoe during the syste () woring during ontinuous staying of the proess in state ( i ) THEOREM The inoe funtional N S S for Syste as a whole is a frational-linear funtional onerning the distributions G { G ( u) ( i ) E } ( i) defining the Marov hoogeneous strategy The final stage of the researh is a onstrution of the optiu ontrol strategy For solving of the proble we use the nown fat []: if a frational-linear funtional has an etreu (a aiu or a iniu) this etreu is reahed in a lass of the deterined strategy (fied deterined probability easure P( ) Desribe the set of the deterined strategies To ontrol of the ustoer type strategy is defined by equality (9) Fied deterined easure in state ( il ) is defined by equality hain at hosen fied strategy is deterined; At hosen fied strategy harateristis (8) and (9) are alulated; The inoe funtional (7) is alulated for the hosen strategy; Using all fied strategies and the values of the inoe funtional for these strategies we define the aial inoe and optiu strategy REFERENCES [] Barzilovih E Belyaev Ju Kashtanov VA 983 The probles of atheatial reliability-theory Mosow Radio I Svyaz [2] Jewell WS 967 Marov-renewal prograing // Operation Res 967 No P [3] Jewell WS 967 Controlled Sei-Marov proesses Mosow: Mir [4] Kashtanov VA 2 Controlled sei-arov proesses in odeling of the reliability and redundany aintenane of queueing systes // Coputer Modelling and New Tehnologies Vol 4 No P 26 3 [5] Kashtanov VA Kondrashova EV 2 «Controlled sei-arov queueing odel» ALT 2 (Aelerated Life testing Reliabilitybased Analysis and Design) Universite Blaise Pasal p [6] Kashtanov VA Kondrashova EV 2 «Controlled sei-arov queueing odels» IMAACA 2 International Conferene on Integrated Modeling and Analysis in Applied Control and Autoation Italy Roe p-8 [7] Kashtanov VA Kondrashova EV 22 «Analisys of arrival flow ontrolled with Marov hain Reliability Journal» p [8] Kondrashova EV 22 «Optiization of inoe funtional in ontrolled queueing odel» Large-sale ontrol Journal 36 Mosow: IPO [9] Luantoni D M Hellste K S Neuts M F 99 A single-server queue with server vaations and a lass of non-renewal arrival proesses// Advantage of Applied Probability V 22 C vans WA 994 Approahes to intelligent inforation retrieval Inforation Proessing and Manageent 7 (2) [] Luantoni D M 99 New results on the single server queue with a bath Marovian arrival proess// Couniations in Statistis Stohasti Models V 7() C -46G O Young Syntheti struture of industrial plastis (Boo style with paper title and editor) in Plastis 2nd ed vol 3 J Peters Ed New Yor: MGraw-Hill 964 pp 5 64 G ( ) P{ u( t) / ξ( t) ( i l )} p () il () il (2) G ( n) P{ u( t) n / ξ( t) ( i l )} p ( il) () il n Thus we reeive the algorith for searhing of optiu strategy: For the fied strategy (2) the atri of transition probabilities (3) is alulated; For this atri the syste of the algebrai equations is solved and the stationary distribution of ebedded Marov