BULGARIA ACADEMY OF SCIECES CYBERETICS AD IFORMATIO TECHOLOGIES Volume 8, o Sofa 28 Adaptve Dynamcal Pollng n Wreless etworks Vladmr Vshnevsky, Olga Semenova Insttute for Informaton Transmsson Problems RAS, E-mals: vshn@tp.ru olgasmnv@tp.ru Abstract: We consder a pollng system wth adaptve pollng mechansm descrbng the performance of broadband wreless W-F and WMax networks. A server vsts ueues n cyclc order dependng on the state of ueues n the prevous cycle n the followng way. A ueue s skpped (not vsted) by a server f t was empty n the prevous cycle. The skpped ueues are polled n the next cycle only. Such a pollng mechansm s referred to as adaptve one. We propose to reduce an adaptve pollng mechansm to a Bernoull pollng scheme allowng nvestgaton of the model wth the mean value analyss. Keywords: pollng system, adaptve mechansm, mean value analyss.. Introducton The models of pollng systems whose study dates from the late 95 s found wde use n the publc health systems, ar and ralway transportaton, and communcaton systems. The number of works on the pollng systems s ute large. Classfcaton of pollng systems, methods and results of ther nvestgaton are revewed n [4, 5, 6,, 3, 4]. The rapd development of the telecommuncaton networks and cellular communcatons n partcular whch s often called the wreless revoluton has made t necessary to create and nvestgate the models descrbng the features of such systems and networks []. The pollng models for nvestgatng the characterstcs of personal and local wreless networks are analyzed n [3, 7]; those ntended for the regonal wreless broadband regonal networks wth centralzed control, n [8, 9]. 3
The adaptve pollng mechansm wth ueue skps n a cycle adeuately descrbes the performance of broadband wreless W-F and WMax networks where the number of abonent statons s large. When base staton polls the abonent ones cyclcally t can be mpossble to poll all statons n a cycle thus some of them have to be skpped. One of the crtera to skp (not to poll) a ueue (abonent staton) n a cycle s ts emptness at the prevous pollng moment. The pollng moment of a ueue s referred to as a moment when the server (base staton) checks f there are packets n a ueue to be transmtted. Unfortunately the adaptve mechansm s hard enough to be analyzed so we use approxmaton methods and pollng schemes, e.g. a threshold pollng scheme [2]. In the present paper we show how the adaptve pollng mechansm can be reduced to a Bernoull scheme and develop an approxmate algorthm for calculaton of the mean watng tme n a ueue on the base of mean tme analyss [5]. 2. Model We consder a pollng system wth a sngle server and ueues, 2. Each ueue has nfnte buffer capacty. The server vsts and serves the ueues n a cyclc adaptve order. Such an order s not fxed but changes at the begnnng of a cycle dependng on the states of ueues n the prevous cycle. The -th ueue has ts own Posson nput of customers wth rate λ. The servce tmes n the ueue are ndependent, dentcally dstrbuted random varables wth a mean b and second moment b. Servce at each ueue s a gated one: when the server vsts a ueue t serves all, and only, customers present n the ueue at the pollng nstant. When the server vsts the ueue the setup tme s ncurred of whch the frst and second moment are denoted by g and g, =,. We refer to a ueue pollng nstant as a moment when the server has completed the setup tme and ready to serve the ueue. It s supposed that the server does not know the ueue length untl the setup tme s expred. If a ueue s empty at ts pollng moment the server wll skp (not vst) ths ueue n the next cycle. If all ueues are to be skpped the server ntates an empty cycle,.e. takes a vacaton havng an exponental dstrbuton wth mean τ and then polls all ueues startng from ueue. The occupaton rate ρ at ueue s defned as ρ =λ b, =,. The total occupaton rate s ρ = = ρ. The necessary and suffcent condton for the stablty of the pollng system under consderaton s ρ < [2]. In the next secton we develop the approxmate approach and reduce the adaptve mechansm to Bernoull pollng, []. The system under consderaton dffers from the one n [] by the fact that the setup tme s ncurred only f a ueue s vsted for servce. 4
3. Mean cycle length and probablty of a ueue vst Suppose that the ueue s vsted n the current cycle wth a probablty u. Suppose that the probablty does not depend on the number of the cycle. In that case the adaptve pollng mechansm can be approxmated by a Bernoull scheme whch s descrbed as follows. The set of probabltes (u,, u ), s fxed, < u, =,. The ueue s served n the cycle wth a probablty u and wth addtonal probablty the server moves to the next ueue. For the adaptve mechansm the probabltes u,, u depend on the mean cycle length C and can be calculated as u ( C = u + u e λ ), where C s the mean cycle length. The cycle length means the tme for the server to vst ueues from to excludng ueues to be skpped. Let us gve a short explanaton for the formula above. A ueue s vsted n a cycle when t was skpped n the prevous cycle (wth a probablty u ) or t was vsted n the prevous cycle (wth a probablty u ) and customers arrved to the ueue durng the ntervst tme (the tme between two successve vsts to the ueue). It follows from the euaton above that () u =, =,. C + e λ The mean ueue length s determned by the formula g ( ) u + τ u = = C =. ρ The relatons () and gve the system of euatons for calculaton of the unknown values C and u,, u. The second way to determne the probabltes u, =,, can be appled when the probablty that a ueue s empty at a pollng nstant can be calculated or estmated. Ths way s descrbed as follows. Consder the stochastc process c,, where c s the status of the ueue n -th cycle, that s c = f the ueue s skpped and c = otherwse. () The state of the process c,, depends on ts prevous state and the ueue () state n the -th cycle. If c = and the ueue s empty at the pollng nstant n () () the ( )-th cycle, we have c =. Otherwse, c =. The probablty u that the ueue s vsted by the server n an arbtrary cycle s the statonary state probablty () that c =, u = P c =, =,. () lm { } () Let π be the statonary state probablty that the ueue s empty at a pollng nstant and x () lk, lk, =,, be one step transton probabltes of the process () { c, }, 5
() () (3) x =, x =, () () () () (4) x = π, x = π. The probablty u can be calculated from the balance euaton () () () () u = P{ c = } x + P{ c = } x. Hence, () u = ( u) + u( π ) and from (3) we have (5) u = () π. + ote that probabltes () π, =,, are unknown and the formula (5) can only be exploted when these probabltes are calculated or estmated. 4. Mean ueue length In ths secton we derve the approxmaton for the mean ueue length at an arbtrary tme on the base of mean value analyss [5]. Let be the average tme the server spends n the ueue plus the average setup tme to ueue + under condton that the ueue + s vsted by the server, =,. We suppose that n the empty cycle the server s cyclcally vstng all the ueues and t spends the mean tme τ / n each ueue wthout customer servce. The value s defned as = ρc+ g+ u+ + vτ /, =,, where v= ( u ) = s the probablty that a cycle s empty, I { = } euals f = and euals otherwse. As n [5] we defne the ( )-perod as the sum of consecutve vst tmes startng from ueue, the mean of the perod s defned as + =, =,. The fracton of tme the system spends n the ( )-perod s gven by, n =, =,. C The mean of a resdual ( )-perod s gven by R =, =,, 2 where, s the second moment of ( )-perod length. Denote by L the mean ueue length at an arbtrary epoch of vstng the ueue, =,. The correspondng uncondtonal ueue length s defned as 6
L = L, =,. n, n The value L n the case = s the sum of two varables L and L. The value L s the number of customers to be served at an arbtrary epoch of vst to ueue. The value L, s the number of customers that arrved durng the servce tme of the ueue and wll be served n the next cycle. In case L = L. That s ρ L = L I{ = } + L, =,. u The correspondng uncondtonal mean ueue length L s calculated as ρ ρ (6) L = L + L = nl, n+ L, =,. u u One more euaton for the value L can be derved by Lttle s law L = λw, (7) where W s the mean watng tme n the ueue (the tme from a customer s arrval at ueue untl ts servce starts). The customer arrvng to ueue has to wat for the servce of all customers L watng before the gate on ts arrval. Further, t has to wat untl the frst pollng nstant of ueue euallng a resdual ( )-perod,.e., a resdual cycle. And wth probablty u the ueue was not vsted n the prevous cycle, so the customer has to wat one more cycle. Thus, the mean watng tme W s gven by (8) W = L b + R ( u ),, + C, =, whch, n combnaton wth Lttle s Law (7), gves us the followng relaton (9) L = L + λr + λ ( ) u C, =,., The number of customers at an arbtrary moment of the perod ( ) s the number of Posson arrvals durng the age of the perod plus the arrvals durng the cycle f the ueue was not vsted n the prevous cycle, + n, () L n, = λr + λ ( ) u C, =,., Substtutng (6) n (9) we get ρ () ( ρ) n L, n, + L = λr + λ ( ) u C, =,., u ote that euatons () and () form the system of (+) lnear euatons for L, L and R,. To calculate the unknown mean resdual ( )-perods from ths system, below we obtan dependence of R, on L and L. 7
The mean resdual ( )-perod lasts at least the sum of the servce tmes of the ρc customers behnd the gate f the ueue s vsted for servce. Wth probablty the mean resdual servce tme b Rb = 2b and the mean setup tme for ueue + s added gven that the ueue + s not u s skpped. Further, wth a probablty the mean resdual setup tme for the ueue + 8 R g+ + + g = 2g s generated. Fnally, the mean resdual ( )-perod euals to the mean resdual tme server spends at ueue that s, τ / f the cycle s empty (wth probablty v). Thus, we have ρc u+ g+ vτ R = ub L + Rb+ u g R, + + + g +. +, Consder the case of the mean resdual ( 2)-perod. Wth probablty,, the, 2 value of R, euals R 2, + g + 2u+ 2 plus the servce tmes of customers present n the ueue + at an arbtrary moment when the server vsts the ueue and of customers arrvng to the ueue + durng the mean tme R, gven that the, t euals. Thus, ueue + s vsted. Wth addtonal probablty (,, 2 ) (3) + + R = R + u s + R + b u + 2 2 2 ( ) λ L,, + + + +, + + 2 + R = ( R ( ) + ρ u +,, + + + 2 2 + u+ 2s+ 2 + L +, b+ u+ ) + R +,, =,. 2 The values R, for = 2, can be obtaned n a smlar way,, (4) R = R ( ) ρ nu,, n + + + + + u+ n+ s+ n+ + L + n, b+ nu+ n ( ρ+ mu+ m) + + m= n+ + R 2 +,, =,, =,. R +,
Fnally, the euatons -(4) form a set of 2 lnear euatons. Solvng the euatons ()-() and -(4), we get the unknowns L, and. Then, the uncondtonal mean ueue lengths and mean delays are easly calculated from (6) and (8). 5. umercal example To llustrate the obtaned results we present numercal examples. We compare the approxmate results presented above wth smulaton results. Let us consder a symmetrc pollng system wth two ueues and exponentally dstrbuted servce tmes. In ths case we omt the subscrpt for the ueue characterstcs. The mean servce tme b =.3, mean setup tme g =.9. The approxmate results (column T ), smulaton results (column E ) and relatve error of comparson (column ) are shown n Table. We compare the mean cycle length C, probablty u that ueue s polled n the cycle and mean ueue length L. Table. A symmetrc system wth two ueues λ, ρ C, u, L T E, % T E, % τ =.5 τ =. C λ =.5,.54.57 2.6.7.68.77 ρ =.3 u.526.534.5.528.537.7 L.24.233 2.9.247.235 4.9 τ =.5 τ =. C λ =,.3.3 3.28.326.328.6 ρ =.622 u.595.596.6.6.62.33 L.697.74 5.98.723.744 2.86 ow let the number of ueues n the system be 5. The nput ntenstes are λ =, λ 2 = 2, λ 3 =.5, λ 4 = 6, λ 5 =.5. The mean setup tme s the same for all ueues and euals.5, the mean tme of an empty cycle τ =.5. Table 2 shows results for two values of the mean servce tme.5 and.7. Table 2. A system wth fve ueues L T E, % T E, % C, u, L b =.5, ρ =.5 b =.7, ρ =.7 C.32.33 3..587.593.5 u.579.62 6.74.642.678 5.33 u 2.654.696 6.7.764.769.7 u 3.539.568 5.25.572.68 5.98 u 4.87.85 2.25.974.9 6.5 u 5.539.568 5.32.572.67 5.9 L.434.429.5.763.738 3.33 L 2.843.793 5.76.47.4 4.87 L 3.222.227 2.3.393.4.76 L 4 2.5 2.35 6.8 5.3 4.772 4.96 L 5.222.229 3..393.45 2. R, 9
The results obtaned for the mean servce tmes b =.7, b 2 =.5, b 3 =., b 4 =.25, b 5 =.4 are shown n Table 3. Table 3. onsymmetrc servce n ueues C, u, L T E, % C.32.324.89 u.579.62 5.44 u 2.654.676 3.2 u 3.539.564 4.38 u 4.87.89 6.9 u 5.539.566 4.74 L.473.58 6.88 L 2.846.887 4.73 L 3.287.285.7 L 4 2.46 2.376.66 L 5.287.286.35 6. Concluson A pollng system wth adaptve pollng mechansm s consdered. The adaptve mechansm means that the order n whch the server vsts ueues depends on the states of ueues n the prevous cycle,.e. the server does not vst ueues that were empty at ther pollng moments n the prevous cycle. The adaptve mechansm s reduced to a Bernoull one, that s a ueue s polled n a cycle wth some probablty. The mean watng tme n each ueue s obtaned on the base of mean value analyss. R e f e r e n c e s. A l t m a n, E., U. Y e c h a l. Cyclc Bernoull Pollng. ZOR Methods and Models n Operatons Research, Vol. 38, 993, o, 55-76. 2. F r c k e r, C., M. R. J a b. Monotoncty and Stablty of Perodc Pollng Models. Queueng Systems, Vol. 5, 994, o 3, 2-238. 3. M o r a n d D., A. Z a n e l l a, G. P e r o b o n. Performance Evaluaton of Bluetooth Pollng Schemes: An Analytcal Approach. ACM Moble etworks Applcatons, Vol. 9, 24, o 2, 63-72. 4. L e v y, H., M. S d. Pollng Systems: Applcatons, Modelng and Optmzaton. IEEE Transactons and Communcatons, Vol. 38, 99, o, 75-76. 5. T a k a g H. Queueng Analyss of Pollng Models: An Update. In: Stochastc Analyss of Computer and Communcaton Systems. H. Takag Ed. Amsterdam, orth-holland, 99, 267-38. 6. T a k a g H. Queueng Analyss of Pollng Models: Progress n 99-994. In: Fronters n Queueng. J. H. Dshalalow, Ed. CRC, Boca Raton, FL, 997, 9-46. 7. Vshnevsky, V. M., A. I. Lyakhov. Adaptve Features of IEEE 82. Protocol: Utlzaton, Tunng and Modfcatons. In: Proc. of 8th HP-OVUA Conf., Berln, June 2. 8. Vshnevsky, V. M., A. I. Lyakhov,.. Guzakov. An Adaptve Pollng Strategy for IEEE 82. PCF. In: Proc. of 7th Int. Symp. on Wreless Personal Multmeda Communcatons (WPMC 4). Vol.. Abano Terme, Italy, September 2-5, 24, 87-9.
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