Elevator-Type Polling Systems

Transcription

1 Elevator-Type Pollng Systems Ruth Shoham Ur Yechal Department of Statstcs, School of Mathematcal Scence Raymond and Beverly Sackler Faculty of Exact Scences Tel Avv Unversty, Tel Avv 69978, Israel November 1992 Abstract The basc pollng system s a confguraton of N queues attended by a sngle server, usually n a cyclc order. In an Elevator-type pollng scheme, nstead of movng n a cyclc fashon, the server scans the channels back and forth, resdng n each queue for a duraton of tme determned by the servce dscplne. Such systems have a wde varety of applcatons n the areas of telecommuncatons, computer networks, manufacturng, mantenance and repar, etc. In ths work we apply a unfed approach to study four Elevatortype pollng systems, dstngushed by ther servce regmes. The system-models are called Elevator Exhaustve, Elevator Gated, Elevator Globally-Quas-Exhaustve and Elevator Globally-Gated the Exhaustve, Gated and Globally-Gated dscplnes were partally studed n the lterature). For each system we provde a comprehensve analyses regardng cycle tmes n the and down drectons, server s sojourn tmes n the varous channels n each drecton, customers watng tmes, and channels queue szes. Furthermore, we derve condtons under whch the duratons of the and down cycles are the same. The calculaton of customers mean watng tmes are based on a dervaton of a general relaton-formula that can be used n conjuncton wth many other pollng schemes. Sported by a Grant from the France-Israel Scentfc Cooperaton n Computer Scence and Engneerng) between the French Mnstry of Research and Technology and the Israel Mnstry of Scence and Technology, Grant Number

2 1 Introducton Queueng systems consstng of N queues channels) served by a sngle server whch ncurs swtch-over perods when movng from one queue to another have been wdely studed n the lterature and used as a central model for the analyss of a wde varety of applcatons n the areas of telecommuncatons, computer networks, manufacturng, etc. Very often such applcatons are modeled as a pollng system n whch the server vsts the queues n a cyclc order or accordng to some pollng table c.f. Baker & Rubn 1987]). In many of these applcatons, as well as n most pollng models, t s common to control the amount of servce gven to each queue durng the server s vst. Two wdely used polces are the Exhaustve and the Gated regmes, whose analyss for queues wth nfnte buffers) has been extensvely studed n the lterature see Takag 1986] and 1990]). Recently, the Globally-Gated and the Globally-Quas-Exhaustve servce regmes were proposed by Boxma, Levy & Yechal 1992], who provded a thorough analyss of the cyclc Globally-Gated scheme. Boxma, Westsrate and Yechal 1993] further extended the Globally-Gated model to nclude server nterrtons, and appled t to a real-world reparman problem. A Globally-type regme uses a tme-stamp mechansm for ts cyclc reservaton: the server performs a Hamltonan tour through the queues, and uses the nstant of cycle-begnnng as a reference pont of tme. If n 1, n 2,..., n N ) s the statevector of the number of customers present at the varous queues at the start of the Hamltonan tour when all gates are globally closed) then, under the Globally-Gated regme, the server serves exactly n customers at queue, whereas under the Globally-Quas-Exhaustve dscplne, the server resdes n queue for the duraton of n ordnary busy perods. In ths paper we concentrate on the Elevator-Type scan) pollng scheme: nstead of movng cyclcally through the channels, the server frst serves the channels n one drecton,.e. n the order 1, 2,..., N cycle) and then reverses ts orentaton and serves the channels n the opposte drecton down cycle),.e. gong through channels N, N 1,..., 1. Then t changes drecton agan, and keeps movng n ths manner back and forth. Ths type of pollng mechansm s encountered n many applcatons, e.g. t models a common scheme of addressng a hard dsk for wrtng or readng) nformaton on or from) dfferent tracks. The Elevator scheme saves the return walkng tme from channel N to channel 1 when compared to cyclc pollng systems). 2

3 Addtonal motvaton for consderng Elevator-type models emerges from the results obtaned by Browne & Yechal 1989] when studyng Dynamc Prorty Rules for Cyclc-Type Queues. In that paper t s shown that the total tme to complete a Hamltonan tour of the N channels, when the vst order s gven by the permutaton π = π 1), π 2),..., π N) ), s N EC) = a π) 1 + α πr) ), 1) =1 r=+1 where a s the ntal core of work at channel, and α s the growth rate of work at that channel. Ths expresson s mnmzed by followng the permutaton based on orderng the statons n ncreasng values of the ndex a α. Moreover, f a two-cycle horzon s consdered, then, n expectaton, the length of ths horzon s mnmzed f the server performs the second Hamltonan tour n the reverse order of the frst tour. Ths mples an Elevator type pollng scheme see also Yechal 1991]). Elevator or scan) type pollng systems have been already studed n the lterature. Coffman and Hofr 1982] analyzed the Exhaustve servce regme assumng constant servce and swtch-over seek) tmes. They derved the Probablty-Generatng-Functons PGFs) of the number of customers packets) at the varous queues at pollng nstants, as well as at swtch-over tmes. Expressons for customers mean watng tmes are then calculated. Swartz 1982] consdered a slotted-tme model under the Gated servce dscplne and obtaned the PGF of the state of the system at pollng nstants. Takag & Murata 1986] further analyzed scan-type TDM and pollng systems under both the Exhaustve and Gated servce regmes. They computed the mean delay values of requests at the varous statons and show that a dscrmnaton exsts among the statons due to ther relatve postons. Models wth a sngle buffer for each queue has also been studed for a cyclc Hamltonan) type pollng scheme Browne & Yechal 1991]), as well as for heterogeneous scan pollng procedure Bunday, Sztrk & Tapsr 1992]). Altman, Khamsy & Yechal 1992] ntroduced and studed the Elevator Globally-Gated scheme. A surprsng result for that system s that mean watng tmes at all statons are the same. In ths paper we employ a unfed approach to study, analyze and extend results for four servce regmes under the Elevator type pollng scheme. The regmes are the Exhaustve, Gated, Globally-Quas-Exhaustve and Globally- Gated. We derve a new and general formula for calculatng mean watng 3

4 tmes of customers n varous queues n an arbtrary pollng scheme not necessarly of Elevator-type), and use ths formula for calculatng performance measures for each of the servce regmes mentoned above. Furthermore, we calculate the server s sojourn tmes n each queue n the and down drectons, separately, and fnd condtons under whch the and down cycles are equal. For the Globally-Gated regme we reestablsh the surprsng result Altman, Khamsy & Yechal 1992]) that all mean watng tmes are equal. Ths s the only known non-symmetrc pollng scheme that acheves such a farness phenomenon. The structure of the paper s as follows: In Secton 2 a general descrpton of Elevator-type procedures ndependent of the servce dscplne n each channel) s presented. In Secton 3 we derve a general formula for calculatng mean watng tmes of customers n the varous channels. In Sectons 4, 5, 6 and 7 we provde analyses for the Elevator Exhaustve, Elevator Gated, Elevator-Quas-Exhaustve and Elevator Globally-Gated systems, respectvely. 2 The Model We consder a pollng system consstng of a sngle server and N ndependent nfnte-buffer queues channels). Customers arrve at queue = 1, 2,..., N) accordng to a Posson process wth rate λ. The server moves from one channel to another n an elevator scan) fashon: t frst serves the channels n one drecton,.e. n the order 1, 2,..., N drecton), and then reverses ts orentaton and serves the statons n the opposte down ) drecton, gong through statons N, N 1,..., 1. The server stays at channel = 1, 2,..., N) for a length of tme determned by the servce dscplne and then moves to channel + 1 or 1, accordng to the pollng orentaton. It keeps movng from one channel to another even when there are no customers n the system. Each customer n channel carres an ndependent random servce requrement dstrbuted as V and havng dstrbuton functon G.). The flow-rate of work to channel s ρ = λ EV ), and the total flow-rate of work nto the system s ρ = N =1 ρ. When leavng channel and before movng to the next channel the server ncurs a swtch-over walkng tme) perod, the duraton of whch s a random 4

5 varable. Ths varable denotes the swtch-over tme from channel to channel + 1 n the drecton, and from channel + 1 to channel n the down drecton θ and θ down, respectvely). In varous applcatons t s common to assume that all swtch-over tmes are ndependent, and for each the and down swtchng dstrbutons are the same,.e. θ = θ = θ down. The perod durng whch the server moves down) s called an down ) cycle, and s denoted by C 1 C 2 ). A full cycle s C = C 1 + C 2. Fnally, throughout the paper the Laplace-Steltjes-Transform LST) of a random varable X s denoted by Xω) = E{exp ωx)}. 3 A General Result For Mean Watng Tmes A common method for calculatng mean watng tmes ncurred by customers n the varous queues n pollng systems s presented n Takag 1986], where specfc calculatons for the cyclc Exhaustve and the cyclc Gated regmes are performed. The method s based on obtanng a set of mplct equatons for the PGFs of the number of customers found at the varous channels at pollng nstants, dfferentatng these PGFs and dervng a set of lnear equatons whose soluton enables one to calculate the desred mean watng tmes. We develop an alternatve method for obtanng mean watng tmes n an arbtrary pollng system. The method s based on a dervaton of a general equaton for EW ), the mean watng tme of customer n queue, whch s then used for each servce regme accordng to ts specfc features. The mean watng tmes n all Elevator-Type schemes mentoned above are derved wth the ad of ths equaton by calculatng the watng tme n the cycle and n the down cycle, separately. The mean watng tme s then gven by a weghted sum of the two means, where the weghts are the probabltes of fndng the server n the cycle or n the down cycle, respectvely. Consder the probablty generatng functon, Q z) = Ez L ), of the number of customers, L, left behnd by an arbtrary departng customer from channel. As the dstrbuton of number of customers n the system at epochs of arrval and epochs of departure are dentcal, then by the well known PASTA phenomenon Posson Arrvals See Tme Averages), Q z) also stands for the generatng functon of the number of customers at channel n a steady state regme at an arbtrary pont of tme. 5

6 Consder the system n steady-state. Let T be the total number of customers served n channel durng a vst of the server to that channel, and let L n) n = 1, 2,..., T ), be the sequence of random varables denotng the number of customers that the n-th departng customer from channel countng from the moment that the channel was last polled) leaves behnd hm. Then the PGF s gven see Takag 1986], p. 78) by Q z) = E T n=1 z ) L n) 2) E T ) Let X denote the number of customers present at channel at ts pollng nstant, and denote by V n) the total servce tme of n customers n channel. Also let A t) be the number of Posson arrvals to channel durng a tme nterval of length t. Then, L n) = X n + A V n)). Thus, the evaluaton of the expresson for Q z) becomes: Q z) = 1 E T ) E = 1 E T ) E z X T z X n+a V n)) = 1 n=1 E T ) E T z X z n+a V n)) n=1 T z n e λ V n)1 z) = 1 ] T n n=1 E T ) E Ṽ λ z X 1 z)) n=1 z ] T = 1 E T ) E Ṽ z X λ 1 z)) z 1 Ṽ λ 1 z)) z 1 Ṽλ 1 z)) = 1 E T ) 1 z Ṽλ 1 z)) E z X T Ṽ λ 1 z)) z T Ṽ λ 1 z)) ] )] T 3) The average number of customers at channel at an arbtrary pont of tme s gven by: EL ) = Q z) z = ET 2 ) ET ) z=1 2ET ) z 1 + ρ ) + EX T ) ET 2 ) + ρ 4) 2ET ) Let W denote the watng tme of an arbtrary customer at queue. The Laplace Steltjes Transform LST) of W and ts expectaton are obtaned usng the well known relatons: W λ 1 z))ṽλ 1 z)) = Q z) 6

7 λ EW ) + λ EV ) = EL ) 5) Thus, the average watng tme for an arbtrary customer at channel s gven by EW ) = ET 2 ) ET ) 1 + ρ ) + EX T ) ET 2 ) 2λ ET ) 2λ ET ) 6) The followng theorem gves some nsght nto the result gven by Eq. 6). Theorem 1 Let A = A H + R ) and X = A H ) be two Posson random varables each wth ntensty λ ), representng the number of customers that have arrved to channel durng some random perods H + R and H, respectvely. Then EA 2 ) EA ) 2EA ) EX A ) EA 2 ) 2EA ) E H + R ) 2 ] = λ 2E H + R ) = λ EH R ) + λ ER 2 ) + ER ) 2EH + R ) Proof: As A s Posson, EA ) = λ EH + R ) EA 2 ) = E A H + R )) 2] = E H,R E A H + R )) 2]] = E H,R λ 2 H + R ) 2 + λ H + R ) ] = λ 2 E H + R ) 2] + λ E H + R ] Thus, EA2 ) EA ) EH +R ) 2EA = λ 2 ] ) 2EH +R. Also, as A ] H + R ) = A H ) + A R ), we have { EX A ) = E H,R E A H )) 2 + A H )A R ) ]} ] = E H,R λ 2 H 2 + λ H + λ 2 H R 7) 8) Thus, = λ 2 EH 2 ) + λ EH ) + λ 2 EH R ) EX A ) EA 2 ) 2EA ) = λ EH R ) + λ ER 2 ) + ER ) 2EH + R ) 7

8 Spose that n some pollng systems the number of customers present n channel at ts pollng nstant, X, s the number of Posson arrvals to that channel durng some random tme H,.e X = A H ), and T, the total number of customers served durng a vst to channel s gven by A H +R ). Then, t follows from Theorem 1 that EW ) = E H + R ) 2 ] 1 + ρ ) λ EH R ) + λ ER 2 ) + ER ) 2E H + R ] 2λ E H + R ] That s, the mean watng tme s comprsed of three elements, the frst two of whch are 9) ) The mean resdual tme of the random tme perod H + R gven by αh, R) EH +R ) 2 ] 2EH +R ] ). ) The servce tme requred by customers who have arrved at that channel durng the past part of the perod H + R, but before the arrval of the specfc customer gven by ρ αh, R) ). The last term n Eq. 9) s due to the dependence of the random varables X and T. 4 Elevator Exhaustve Scheme In ths secton we analyze the Elevator-pollng Exhaustve-servce scheme. We calculate expressons for the LST and means of the sojourn tmes of the server n varous queues n the and down cycles, and derve explct expressons for the mean duratons of the and down cycles. Fnally, we obtan formulae for calculatng mean watng tmes of arbtrary customers n the varous queues, both n the and down drectons. 4.1 Cycle Tmes Spose that at tme 0 the state of the system s n 1, n 2,..., n N ), where n s the number of customers present n channel. The server than starts ts cycle servng each channel untl t s empty, and then moves on to the next channel. Upon completon servce at channel N the server starts ts down 8

9 movement at that moment, because of the Exhaustve servce dscplne, channel N s empty). The down cycle s completed when channel 1 s exhausted. A new full cycle then starts agan. It s thus clear that n 1 = n N = 0 n every future cycle. That s, the server vsts channels 1 and N only once n each full cycle, whereas channels 2,..., N 1 are each vsted twce. Let Y 1) and Y 2) be the occaton tme of the server n channel n the and down drectons, respectvely. Then, Y 1) = n A S 1) B j + 1) B j + θ, = 1, 2,..., N 10) where {B j } s a sequence of..d random varables all dstrbuted as an M/G /1-type busy perod wth mean EB ) = EV )/1 ρ ), second moment EB 2 ) = EV 2 )/1 ρ ) 3 and LST B ω). A t) s a Posson random varable wth rate λ, countng the number of arrvals to channel durng a tme perod of length t. S 1) 1 = 1 Y 1) j s the entrance tme to channel. The LST of s thus gven by Y 1) Ỹ 1) ω) = B ω) ] n S1) 1 λ 1 B ω)) ) θ ω), = 1, 2,..., N. 11) Eq. 11) leads to EY 1) ) = n EV ) + ρ 1 EY 1) j ) + Eθ ), = 1, 2,..., N 12) 1 ρ 1 ρ where ρ = λ EV ) s the amount of work flowng to channel per unt tme. Usng the defnton Z 1) = ES 1) ) and addng Z 1) 1 to both sdes of Eq. 12), we obtan a set of dfference equatons n Z 1) Z 1) 1 1 ρ Z 1) 1 = n EV ) 1 ρ + Eθ ), = 1, 2,..., N ; Z 1) 0 = 0. 13) The soluton of Eq. 13) s Z 1) = nj EV j ) + 1 ρ j )Eθ ] j ) 1 + ρ r ), = 1, 2,..., N. 1 ρ j r=j+1 1 ρ r 14) 9,

10 The mean duraton of the cycle s gven by substtutng = N n Eq. 14): EC 1 ) = Z 1) N = nj EV j ) + 1 ρ j )Eθ ] j ) N 1 ρ j 1 + ρ r ), 15) r=j+1 1 ρ r Observng the system at the end of the cycle, the system s state s A 1 θ 1 + S 1) N S 1) 1 ), A 2 θ 2 + S 1) N S 1) 2 ),..., A N 1 θ N 1 + S 1) N S N 1), 1) 0). The length of tme requred by the server to move back from channel N to channel s N 1 r=0 Y N r. 2) Hence, Y 2) = wth LST Ỹ 2) A θ +S 1) N S1) N 1 + r=0 = A θ ω) = E exp λ 1 B ω)) ) Y 2) N r B j ) N 1) + Y r=+1 r +Y r 2) ) B j The mean value of Y 2) s EY 2) ) = λ EB ) Y 1) r=+1 r=+1 r + Y 2) + θ down 1 + θ down 1, = 1, 2,..., N 16) r ) θ λ 1 B ω)) ) θdown 1 ω), = 1, 2,..., N. 17) EY 1) r ) + EY r 2) ) ) + Eθ ) + Eθ down 1 ), = 1, 2,..., N. 18) By addng Z 2) +1 = N r=+1 EY 2) r ) to both sdes of Eq. 18) we obtan a system of dfference equatons Z 2) 1 Z 2) +1 = ρ 1 ρ 1 ρ r=+1 10 EY r 1) ) + Eθ ) + Eθ down 1 ),

11 whose soluton s Z 2) = ρ j j= Nr=j+1 EY 1) r = 1, 2,..., N ; Z 2) N = 0 19) ) + Eθ j ) ] + 1 ρ j )Eθ down 1 ρ j j 1 ) j 1 r= 1+ ρ r 1 ρ r ), = 1, 2,..., N. 20) Now, the mean down cycle tme s gven by substtutng = 1 n Eq. 20): EC 2 ) = Z 2) ρ j 1 = Nr=j+1 EY 1) r ) + Eθ j ) ] + 1 ρ j )Eθ down 1 ρ j j 1 ) j 1 r=1 1+ ρ r ) 21) 1 ρ r As EY r 1) ) = Z r 1) Z r 1, 1) Eq. 14) and Eq. 20) determne explctly the 2N values of Z r 1) and Z r 2) r = 1, 2,..., N) for any ntal system-state n 1, n 2,..., n N ). The mean cycle tme s gven by the sum of the mean cycle and the mean down cycle Eq. 15) and Eq. 21) respectvely). In general the expected value of n j EV j ) s ρ j j 1 r=1 EY r 2) ) + Eθj 1 down )], whch s the expected amount of work flowng nto channel j from the moment the server leaves the channel n the down drecton untl t reenters t n ts drecton. Substtutng the above expresson for n j EV j ) n Eq. 15), we express EC 1 ) n terms of EY r 2) ), smlarly to Eq. 21). In order to fnd expressons for Y 1) and Y 2) for an arbtrary cycle, we note that the number of customers present n queue at the begnnng of an cycle s n = A θ down 1 + ) 1 r=1 Y r 2). Thus Eq. 10) s transformed nto wth LST Y 1) = A θ down ) Y r=1 r +Y r 2) ) Ỹ 1) ω) = { E exp λ 1 B 1 ω)) Y 1) r=1 ) B j r + Y 2) 11 r ) ]} + θ, = 1, 2,..., N 22) θ down 1 λ 1 B ω)) ) θ ω),

12 Then 1 EY 1) ) = λ EB ) EY 1) r=1 r ) + EY r 2) ) ) + Eθ down 1 ) = 1, 2,..., N. 23) ] + Eθ ), s = 1, 2,..., N. 24) The server s total mean occaton tme at channel durng a full cycle EY 1) )+EY 2) ) = ρ 1 ρ,j It follows that EY 1) EY 1) j ) + EY 2) j ) ) + Eθ 1 down ) + Eθ ) +Eθ down )+Eθ ), 1 = 1, 2,..., N. 25) ) + EY 2) ) = ρ EC) + Eθ 1 down ) + Eθ ), = 1, 2,..., N. 26) Substtutng Eq. 26) n Eq. 24) yelds EY 1) ) = ρ 1 ρ 1 r=1 Eθ down r 1 ) + Eθr In a smlar manner, usng Eq. 18), EY 2) ) = ρ 1 ρ r=+1 ) + ρ r EC) ) + Eθ down 1 ) Eθ r ) + Eθr 1 down ) + ρ r EC) ) + Eθ ) ] +Eθ ), = 1, 2,..., N. 27) +Eθ 1 down ), = 1, 2,..., N. 28) Clearly the mean down) cycle, EC 1 ) EC 2 )), s gven by summng the 12

13 expressons for EY 1) ) EY 2) )). The total mean cycle tme s derved by summng the expressons from Eq. 26), and s gven, as expected, by N=1 Eθ down 1 ) + Eθ ) ] EC) = 1 N 29) =1 ρ See Watson 1984] for a general result, and Takag & Murata 1986] for a slotted model). 4.2 Comparson between the and down cycles We are nterested n fndng condtons under whch the means of the and the down cycles are equal. We have Proposton 4.1 In the Elevator-Exhaustve model, EC 1 ) = EC 2 ) whenever θ = θn down and ρ = ρ N +1 = 1,..., N). Proof: Usng Eq. 27) and Eq. 28) = =1 =1 ρ 1 ρ ρ 1 ρ EC 1 ) EC 2 ) = 1 r=1 r=+1 Eθ down =1 r 1 ) + Eθ r 1) EY ) EY 2) ) ) ) + ρ r EC) ) + Eθ down Eθ r ) + Eθr 1 down ) + ρ r EC) ) + Eθ 1 ) ) ] 30) Substtutng θ = θn down and ρ = ρ N +1 for = 1,..., N, n the second term of Eq. 30), settng k = N + 1, and usng N =1 ρ Eθ )/1 ρ ) = N=1 ρ Eθ down )/1 ρ ), we get ] EC 1 ) EC 2 ) = =1 ρ N +1 =1 1 ρ N +1 ρ 1 ρ N k=1 1 r=1 Eθ down r 1 ) + Eθr ) + ρ r EC) ) + Eθ down Eθ down k 1 ) + Eθ k ) + ρ kec) ) + Eθ down 1 ) Substtutng j = N + 1, we readly have EC 1 ) = EC 2 ). 1 ) ] 13

14 4.3 Mean Watng Tmes The mean watng tme n the Elevator-Exhaustve system can be calculated utlzng Eq. 6) derved n Secton 3. Clearly, the total mean watng tme n channel s gven by a weghted sum of the mean watng tmes n the cycle, EW ), and the down cycle, EW down). Therefore, EW ) = 1 EC) EC 1)EW ) + EC 2 )EW down)] 31) To calculate EW ) we recall that the number of customers found by the server at channel at pollng nstant of that channel n the drecton, X), s the number of customers that have arrved at channel durng the tme nterval startng from the moment the server fnshed servng channel n the down drecton, untl t frst returns to that channel n the drecton. That s, X ) = A θ 1 down 1 + r=1 ) ) Y 2) r + Y r 1) The number of customers served at channel n the drecton, denoted by T ), s gven by X T ) = X) ) + A B j Thus, the watng tme n the drecton n the Elevator-Exhaustve system s gven by Eq. 6) wth X) and T ) replacng X and T, respectvely. In a smlar manner, denotng by Xdown), the number of customers found by the server at channel when polled n the down drecton, and by T down) the number of customers served at channel n the down drecton, we have X down) = A θ + T down) = X down) + A ) Y 1) r + Y r 2) r=+1 X down) B j Agan, the watng tme n the down drecton s gven by substtutng X down) and T down) n Eq.6). 14

15 Usng X and T nstead of X ) and T ), respectvely, we have X ET ) = E X + A ET 2 ) = E B j 2 X X + A B j = E X) 2] +2 ρ E X) 2 ] = E ) 2 X) 2] ρ +EX 1 ρ ) and = EX)+λ EB )EX) = EX ) 1 ρ +λ 2 EX 1 ρ )V arb )+E X) 2] λ 2 EB )] 2 + ρ EX) 1 ρ ] ρ λ 2 V arb ) + 1 ρ X EXT ) = E X X + A B j = E X) 2] +λ EB )E X) 2] = E X) 2] 1 ρ Substtutng n Eq. 6), we derve, EW ) = E X)) 2 ] λ 2 EB )) 2 + EX)) λ 2 V arb ) 1] 1 ρ 2 2λ EX)) ) + E X)) 2 ] EX )). 32) 2λ EX)) EW down) s obtaned smlarly wth X down) replacng X ) n Eq. 32). In order to complete the evaluaton of EW ) as gven by Eq.31)), t s just left to calculate EX ) and E X ) 2]. 15

16 4.4 Generatng Functons The values of the frst and the second moments of X) and Xdown) wll now be derved from a set of N PGFs as follows. In a smlar way to the method used n Takag 1986], we defne X j ) and X j down) as the number of customers at channel j at pollng nstant of channel at the and down cycle, respectvely. Recall that durng a full cycle the server vsts each of the channels 2, 3,..., N 1, twce once n each drecton), where channels 1 and N are vsted only once. Let F z) and F down z) be the generatng functons descrbng the state-vector of the system at the pollng nstant of channel n the and down drectons, respectvely. That s F z) def = E F down z) def = E N z Xj ) j, = 2,..., N N z Xj down) j, = 1, 2,..., N 1. 33) The evoluton of the state of the system n the drecton s descrbed by X+1) j = X j ) + A j θ ) + A j X ) B j j 34) A θ ) j = Thus, F +1z) = E = θ N z Xj +1 ) j λ j 1 z j ) F z 1,..., z 1, B λ j 1 z j ), z +1,..., z N,,j = 2,..., N 1. 35) 16

17 For = 1 X2) j = X j 1down) + A j θ 1 ) + A j X1 1down) B 1j j 2 A 1 θ 1 ) j = 1 36) Then, F 2 z) = θ 1 λ j 1 z j ) F1 down B1 λ j 1 z j ), z 2,..., z N.,j 1 37) Note that there s no expresson for F ) 1 z), as there s no pollng of channel 1 n the drecton. The evoluton of the state of the system n the down drecton s X 1down) j = XN 1down) j = It follows that N F 1 down z) = E = θ down 1 X j down) + A j θ 1 down ) + A j X down) B j j A θ 1 down ) j = 38) X j N) + A j θ down N 1 ) + A j A N θ down N 1 ) z Xj 1 down) j XN N ) B Nj j N 1 j = N 39) λ j 1 z j ) F down z 1,..., z 1, B λ j 1 z j ), z +1,..., z N,,j = 2,..., N 1. 40) 17

18 FN 1 down down z) = θ N 1 λ j 1 z j ) F N z 1, z 2,..., z N 1, B N 1 N λ j 1 z j ) 41) Results for the correspondng slotted model where obtaned by Takag & Murata 1986] wthout specfc attenton to the end ponts of the cycle channels 1 and N). Let f j) = F z) z j z=1 and f down j) = F down z) z j z=1. Clearly, EX j )) = f j) and EX j down)) = f down j). Takng dervatves, we obtan a set of 2NN 1) equatons n the 2NN 1) unknowns f j) and f down j): f+1j) = f j)+λ j Eθ ) + f )EB )], 2 N 1, j fj+1j) = λ j Eθ j ) f 2 j) = f1 down j)+λ j Eθ 1 ) + f1 down 1)EB 1 ) ] 42) f 1 down j) = f down j)+λ j Eθ down 1 ) + f down )EB ) ], 2 N 1, j fj 1 down j) = λ j Eθj 1 down ) fn 1 down j) = f N j)+λ j Eθ down N 1 ) + f N N)EB N ) ] 43) From Eqs. 42) and 43) t follows, as expected, that f j j) + fj down j) = λ j 1 ρ j )EC). That s, the total number of customers served at channel j durng a full cycle s equal to the number of customers arrvng at that channel whle the server s away. Let j, k) = 2 F z) j, k) = 2 F down z). Then z=1 f z j z k and f z=1 down z j z k the second moments of X j ) and X j down) are gven by E X) 2] = f, ) + f ) E Xdown) 2] = f down, ) + f down ). f, ) and f down, ) are calculated by solvng 2N 2 N 1) equatons n the 2N 2 N 1) unknowns f j, k) and f j, k). For = 2,..., N 1, we use Eq. 35) to get wth some abuse of notaton) 2 F+1z) 2 θ = F θ F θ F z j z k θ 2 F ] z=1 z j z k z k z j z j z k z j z k 18 z=1

19 where 2 F z) = z j z k 2 θ = z j z k f λ j λ k Eθ ) 2 ) j k λ 2 jeθ ) 2 ) j = k )λ j λ k EB 2 ) + λ k f, j)eb ), k)eb ) + f j, k)+ +λ j f f, )λ j λ k EB )) 2 j k f f )λ 2 jeb 2 ) + 2λ j f, j)eb )+ j, j) + f, )λ 2 jeb )) 2 j = k 0 = j, = k θ z k θ z j F z j F z k = λ k Eθ = λ j Eθ )f )f Smlar equatons are derved from Eq. 37) usng F 2 z). In the same manner we use Eqs. 40) and 41) to derve the correspondng set of equatons for f down j, k). Fnally, by substtutng the desred expressons for EX)), E X) 2] n Eq.32) one gets the value of EW ). The correspondng expresson for EW down) s obtaned smlarly. Substtutng these two terms n Eq. 31) yelds the desred result for EW ). j) k) 5 Elevator Gated Scheme In ths secton we analyze the Elevator-pollng Gated-servce scheme. We calculate expressons for the LST and means of the sojourn tmes of the server n each channel durng the and down cycles, separately, and derve explct expressons for the mean duraton of a full cycle. Fnally, we obtan formulae for calculatng mean watng tmes of customers n the varous queues, both n the and down drectons, as well as n general. 19

20 5.1 Cycle Tmes The analyss of the Elevator-Gated system requres only a slght modfcaton of the correspondng analyss of the Elevator-Exhaustve scheme presented n Secton 4. Under the Elevator-Gated scheme the server moves wards along statons 1, 2,..., N, servng n each channel only those customers present on entrance, and then moves n the opposte drecton through statons N, N 1,..., 1. In contrast wth the Elevator-Exhaustve model, the server may leave unserved customers behnd when t exts a channel. Thus, the server vsts all channels ncludng channels 1 and N) twce n every full cycle. Let {V j } denote a sequence of ndependent random servce requrements n channel havng a common dstrbuton functon G.) and LST Ṽ ω). Let Y 1) and Y 2) denote the total occaton tme of the server n channel from the moment that servce begns untl the end of the swtchover to the next channel) n and down drectons, respectvely. Spose that at the start of an cycle the state of the system s n 1, n 2,..., n N ). Then Y 1) s gven by wth LST Y 1) = n A S 1) V j + Ỹ 1) ω) = Ṽ ω) ] n S1) 1 1) λ 1 V j + θ, = 1, 2,..., N, 44) Ṽω)) ) θ ω), = 1, 2,..., N. 45) where S 1) = Y 1) j. The mean value of Y 1) s gven by EY 1) ) = n EV ) + ρ ES 1) 1) + Eθ ), = 1, 2,..., N. 46) Usng the defnton Z 1) = ES 1) ) and addng Z 1) 1 to both sdes of Eq. 46), we obtan a set of dfference equatons n { Z 1) },.e., Z 1) 1 + ρ )Z 1) 1 = n EV ) + Eθ ), = 1, 2,..., N ; Z 1) 0 = 0 47) The system 47) yelds the soluton Z 1) = n j EV j ) + Eθ )] 1 + ρ r ), = 1, 2,..., N. 48) r=j+1 20

21 Thus, the mean value of the cycle s Now, Y 2) = wth LST Ỹ 2) EC 1 ) = Z 1) N = A Y 1) N + r=+1 N n j EV j ) + Eθ )] 1 + ρ r ). 49) r=j+1 ) 1) Y r +Y r 2) ) V j ω) = E exp λ 1 Ṽ ω) ) Y 1) + And mean value EY 2) ) = λ EV ) EY 1) ) + r=+1 + θ down 1, = 1, 2,..., N 50) r=+1 ) Y 1) r + Y r 2) down θ 1 ω), = 1, 2,..., N. 51) EY 1) r ) + EY r 2) ) ) + Eθ down 1 ), Ths leads to Z 2) 1 + ρ )Z 2) +1 = ρ N r= EY r 1) ) + Eθ down) 1 ), = 1, 2,..., N. 52) wth soluton Z 2) = = 1, 2,..., N ; Z 2) N = 0 53) 1) ρj Z N Z 1) ) j 1 + Eθ down j 1 ) ] j ρ r ), = 1, 2,..., N. 54) j= r= The mean value for the down cycle s EC 2 ) = Z 2) 1 = 1) ρj Z N Z 1) ) j 1 + Eθ down j 1 ) ] j ρ r ), 55) r=1 21

22 as f n 2) 1) j = λ j Z N Z j 1) 1) s the number of customers at channel j at the end of the cycle. In order to derve explct expressons for Y 1) and Y 2), we note that n, the number of customers present n queue at the begnnng of an cycle, s the number of customers that have arrved to channel durng the tme nterval startng From the moment the server has last entered channel n ts down drecton, untl the end of the entre down cycle. Therefore, r=1 ) n = A Y r 2). Then, Y 1) = wth LST Ỹ 1) ω) = { E exp A Y 2) 1 + r=1 And mean value EY 1) ) = λ EV ) 2) Y r +Y r 1) ) ) V j λ 1 Ṽ ω) ) Y 2) + EY 2) ) + 1 r=1 + θ = 1, 2,..., N) 56) 1 r=1 )]} ) Y 2) r + Y r 1) θ ω), = 1, 2,..., N. 57) EY 2) r ) + EY r 1) ) ) ] + Eθ ), The total mean occaton tme of the server at channel s = 1, 2,..., N. 58) EY 1) )+EY 2) ) = ρ 1) EY j ) + EY 2) j ) ) + EY 1) ) + EY 2) ) + Eθ down ) + Eθ,j = ρ EC) + Eθ 1 down ) + Eθ ), = 1, 2,..., N. 59) Substtutng the expresson for EY 1) ) + EY 2) ) n Eq. 58) yelds 1 EY 1) ) = ρ Eθ down r 1 ) + Eθr ) + ρ r EC) ) ] + EY 2) ) + Eθ ), r= )

23 The same apples for Y 2) usng Eq. 52)), ) = ρ EY 2) r=+1 = 1, 2,..., N. 60) Eθ down r 1 ) + Eθ r ) + ρ r EC) ) + EY 1) ) + Eθ down 1 ), = 1, 2,..., N. 61) Eqs. 60) and 61) yeld a set of 2N equatons n the 2N unknowns { Y 1), Y 2) }, whose soluton s gven by EY 1) ) = 1 1 ρ 2 +ρ EY 2) ) = 1 1 ρ 2 { r=+1 { 1 +ρ r=1 Eθ ) + ρ 1 r=1 Eθ r ) + Eθ down Eθ 1 down ) + ρ Eθ down r 1 ) + Eθ r ) + ρ r EC) ] 1 ) r 1 ) + ρ r EC) ] + Eθ down N 1 r= Eθ r ) + Eθr 1 down ) + ρ r EC) ] Eθ down r 1 ) + Eθ r ) + ρ r EC) ] ] ]} + Eθ ) 62) 63) The mean total cycle tme, s derved by summng over the occaton tmes n Eq.59) and agan, as expected, s N=1 Eθ down 1 ) + Eθ ) ] EC) = 1 N 64) =1 ρ The mean and down cycles are now calculated by settng EC 1 ) = N=1 EY 1) ) and EC 2 ) = N =1 EY 2) ). 5.2 Comparson between the and down cycles Smlarly to the result obtaned for the Elevator-Exhaustve case Proposton 4.1), we have, Proposton 5.1 In the Elevator-Gated model, EC 1 ) = EC 2 ) whenever θ = θn down and ρ = ρ down N +1 = 1,..., N). 23

24 5.3 Mean Watng Tmes In any Gated servce regme, the number of customers served durng a vst of the server equals the number of customers present at the channel on arrval. That s X = T. Thus, employng Eq. 6) for the classcal cyclc) Gated regme we have EW Cyclc Gated) = E X) 2 ] EX ) 1 + ρ 2λ EX) ). 65) Accordngly, we can use Eq. 65) to calculate separately EW ) and EW down) n the Elevator-Gated scheme. EW ) s then gven by the weghted sum of EW ) and EW down), Eq. 31). The number of customers found by the server at channel at a pollng nstant to that channel n the drecton, X), s the number of customers that have arrved at that channel durng the tme nterval startng from the moment the server has last entered the channel n the down drecton, untl ts frst return n the drecton. That s, X) = A Y 2) 1 ) ) + Y 2) r + Y r 1) Thus, the watng tme n the drecton s gven by Eq.65) wth X) replacng X., In a smlar manner, denotng by Xdown) the number of customers found by the server at channel when polled n the down drecton, we wrte X down) = A Y 1) + r=1 r=+1 ) Y 1) r + Y r 2) so that the watng tme n the down drecton s gven by substtutng X down) n Eq. 65). To complete the evaluaton of EW ) we turn to calculate the terms EX ) and E X ) 2] for both the and down drectons. 24

25 5.4 Generatng Functons Usng the same defntons as n secton 4.4, but wth respect to the Elevator- Gated case, we wrte N F z) def = E z Xj ), = 1,..., N. 66) j The evoluton of the state of the system n the drecton s descrbed by X j X+1) j ) + A j θ ) + A j V X)) ) j = A θ ) + A V X)) ) 67) j = where V n) s the total tme to serve n customers at channel. Thus, N F+1z) = E z Xj +1 ) j = θ λ j 1 z j ) F z 1, z 2,..., z 1, Ṽ λ j 1 z j ), z +1,..., z N, For X 1), = 1,..., N 1. 68) X j 1down) + A j V1 X1down)) ) 1 j 2 X1) j = A 1 V1 X1down)) ) 1 j = 1 69) therefore, 1 z) = F down Ṽ 1 λ j 1 z j ), z 2,..., z N 70) F 1 The PGFs of the system-state n the down drecton are smlarly defned as N F down z) def = E z Xj down) j, = 1, 2,..., N. 71) 25

26 and the evoluton of the states X 1down) j = X j down) + A j θ down 1 ) + A j V X down)) ) j A θ 1 down ) + A V Xdown)) ) j = 72) X j N) + A N VN X XNdown) j N N )) ) j N 1 = A N VN XN N )) ) j = N It follows that N F 1 down z) = E = θ down 1 z Xj 1 down) j 73) λ j 1 z j ) F down z 1,..., z 1, Ṽ λ j 1 z j ), z +1,..., z N, F down N = 2,..., N 74) z) = F z 1, z 2,..., z N 1, ṼN λ j 1 z j ) 75) Let EX j )) = f j) = F down z) z j z=1. N F z) z=1 z j and EX j down)) = f down j) = Takng dervatves of z) and F down z), we get 2N 2 equatons n the 2N 2 unknowns f j) and f down j), j = 1, 2,..., N, as follows f F +1j) = f j)+λ j Eθ ) + f )EV )], 1 N 1, j j+1j) = λ j Eθ j ) + f j j)ev j ) ] f f 1 j) = f down 1 j)+λ j f down 1 1)EV 1 ), j 1 f 1 1) = λ 1 f down 1 1)EV 1 ) f down 1 j) = f down j)+λ j Eθ down ) + f down )EV ) ], 2 N, j )

27 f down j 1 j) = λ j Eθ down j 1 ) + fj down j)ev j ) ] fn down j) = f N j)+λ j f N N)EV N ), j N fn down N) = λ N f N N)EV N ) 77) The second moments of X j ) and X j down) are calculated wth the ad of the second dervatves. Defne f j, k) = 2 F z) z j z k and f z=1 down j, k) = 2 F down z), then, z=1 z j z k E X) 2] = f, ) + f ) E Xdown) 2] = f down, ) + f down )., ) and f down, ) are obtaned by solvng 2N 3 equatons n the 2N 3 varables for f j, k) and f j, k),, j, k = 1,..., N. For = 1, 2,..., N 1 we use Eq. 68) to get f 2 F+1z) = z j z k z=1 where 2 F z) = z j z k 2 θ F + z j z k 2 θ = z j z k θ z k F z j + θ z j λ j λ k Eθ ) 2 ) j k λ 2 jeθ ) 2 ) j = k F z k + θ f )λ j λ k EV 2 ) + λ k f, j)ev ) +λ j f, k)ev ) + f j, k)+ f, )λ j λ k EV )) 2 j k f )λ 2 jev 2 ) + 2λ j f j, j) + f f, j)ev )+, )λ 2 jev )) 2 j = k 2 F z j z k ] z=1 θ z k θ z j F z j F z k = λ k Eθ = λ j Eθ )f )f j) k) 27

28 Smlar equatons are derved from Eq. 70). In the same manner we use Eqs. 74) and 75) to derve the correspondng set of 2N 3 equatons n the unknowns f down j, k). By substtutng the expressons for EX)) and E X) 2] n Eq.65) one calculates EW ). Usng Eq. 65) agan wth respect to the down cycle, the correspondng expresson for EW down) s obtaned. Fnally, EW ) s calculatng va Eq. 31). 6 Elevator Globally-Quas-Exhaustve Scheme In ths secton we study the Elevator-pollng, Globally-Quas-Exhaustve servce regme. We calculate expressons for the mean occaton tme of the server n each channel, and obtan explct expressons for the mean duraton of a cycle. Fnally, we derve expressons for calculatng the mean watng tme of an arbtrary customer n the varous queues, both n the and down drectons, as well as n general. 6.1 Cycle Tmes Spose that when a cycle starts, the state of the system s L 1) = L 1) 1,..., L 1) N ). Then as was descrbed n the Introducton, under the Elevator-pollng, Globally- Quas-Exhaustve servce regme the server moves along statons 1, 2,..., N drecton), stayng at channel for the duraton of L 1) M/G /1-type busy perods. That s, the server makes a global decson at the start of a cycle as to the sojourn tme t s gong to spend at each channel. At the end of the cycle the globally new state of the system s observed, say L 2) = L 2) 1,..., L 2) N ), and the server moves n the opposte down) drecton, through statons N, N 1,..., 1, stayng at channel for the duraton of L 2) busy perods. Ths type of pollng scheme s an extenson of the cyclc pollng Globally- Quas-Exhaustve servce dscplne suggested by Boxma, Levy & Yechal 1992], and dscussed by Moskovtch 1992]. Consderng for a moment the above mentoned cyclc Globally-Quas- Exhaustve dscplne, n whch the server moves from channel to channel n a cyclc fashon. If θ s the swtch-over tme from channel, then t s easly seen that the cycle duraton s unchanged f we alter the order of 28

29 servce of the channels or the order of the swtchng tmes. Ths follows from the fact that the number of busy perods that the server stays n each channel s determned aprory at the begnnng of the cycle, and s therefore ndependent of any change n the order of vsts to the channels. In partcular, f θ N 1 =1 θ = N 1 =1 θ down θ down, then the cycle duraton remans unchanged f the channels are served n the order 1, 2,..., N or N, N 1,..., 1. Thus, the duratons of the cycle and the down cycle n the Elevatorpollng, Globally-Quas-Exhaustve model are the same, and the entre cycle duraton s equal to the sum of two cyclc Globally-Quas-Exhaustve cycles wth zero swtch-over tme from channel 1 to channel N. Formally, to show the equalty of the and the down cycles we wrte E { exp ωc 1 ) L 1)} = E exp ω N 1 L 1) θ + B k Thus, = N 1 =1 θ ω) N =1 B ω) L1) C 1 ω) = E {exp ωc 1 )} = θ ω)g L 1) where G L 1)z) = E { N =1 z L1) }. Smlarly, C 2 ω) = E {exp ωc 2 )} = θ down ω)g L 2) =1 = θ ω) N =1 =1 k=1 B ω) L1). B1 ω), B 2 ω),..., B N ω) ), 78) B1 ω), B 2 ω),..., B N ω) ) 79) Now, f θ = θ down then, probablstcally, L 1) = L 2) and C 1 = C 2 so that all cycles are probablstcly) the same and equal to the cycle duraton, C g, of an equvalent cyclc Globally-Quas-Exhaustve scheme wth zero swtch-over tme from channel 1 to channel N. Hence, n order to fnd the cycle duraton and the sojourn tme of the server at channel n the Elevator-pollng scheme, we frst gve a bref analyss of the cyclc Globally-Quas-Exhaustve model: Let L 1, L 2,..., L N ) be the state-vector of the system at the begnnng of a cycle, where L denotes the number of customers present at channel, then EL ) = λ EB j )EL j ) + Eθ j ), = 1, 2,..., N. 80),j 29

30 where θ s the swtch-over tme requred when movng from channel to the next channel. By addng λ EB )EL ) to both sdes of Eq. 80) we have EL ) 1 + λ EB )] = λ EB j )EL j ) + Eθ j ) λ EC g ), = 1, 2,..., N 81) where C g denotes the cycle tme n the cyclc Globally-Quas-Exhaustve regme. Then, EL ) = λ 1 ρ )EC g ) 82) Let Y denote the occaton tme of the server at channel calculated from the moment the server enters the channel untl the end of the swtch-over tme to the next channel, and let R be the total net servce tme n channel. Then Clearly, EY ) = ER ) + Eθ ) = EB )EL ) + Eθ ) = ρ EC g ) + Eθ ), so that, as expected, = 1, 2,..., N, 83) EC g ) = EY ) = ρ EC g ) + Eθ ) 84) =1 =1 =1 EC g ) = N=1 Eθ ) 1 N =1 ρ 85) Fnally, settng C g as the cycle tme n the cyclc Globally-Quas-Exhaustve regme wth θ N = 0, the mean of a full cycle n the Elevator scheme s EC) = EC 1 ) + EC 2 ) = 2EC g ) = 2 N 1 Eθ j ) 1 N ρ. 86) 6.2 Mean Watng Tmes As C 1 = C 2, EW ) = 0.5 EW ) + EW down)] 87) 30

31 To obtan EW ) and EW down) we use Eq. 6) wth the approprate terms for X and T. Clearly, T ) = L 1) + A M L 1) ) ) where M L) s the duraton of L M/G /1 type busy perods all dstrbuted as B. X), the number of customers at staton at pollng nstant n the drecton, equals L 1) plus the number of arrvals to channel From the begnnng of the cycle untl the entrance tme of the server. That s, X) = L 1) 1 + A Y 1) j = L 1) 1 + A θ j ) + A Mj L 1) j ) )) Hence, ET )) = EL1) ) 1 ρ 88) ET ) 2 ) = E L 1) ) 2] ) 2 1 +EL 1) ) λ 2 V arb ) + ρ ] 1 ρ 1 ρ 89) E X )T ) ) = E = E = E L 1) 1 + L 1) A θ j ) + A Mj L 1) ] 2 1) + L A M L 1) +A M L 1) ) )) + L 1) j ) )) L 1) + A M L 1) 1 A θ j ) + L 1) 1 ) ) 1 A θ j ) + A M L 1) ) ) 1 A Mj L 1) L 1) ) 2] +λ EB )E L 1) ) 2] +λ EL 1) 1 +λ EB j )EL 1) L 1) 1 ) j ) + λ 2 1 EB ) 31 Eθ j )+λ 2 EL 1) ) )) A Mj L 1) j ) ) j ) ) EB j )EL 1) L 1) j ) 1 )EB ) Eθ j )

32 Thus, E X )T ) ) = E L 1) ) 2] 1 ρ +λ 1 1 ρ ) 1 Substtutng results 88), 89) and 90) n Eq. 6), we obtan EW ) = + Eθ j )EL 1) ) + EL 1) L 1) j )EB j ) ] 2] E L 1) ) + EL 1) 1 ρ ) 2 ) ] λ 2 V arb ) + ρ EL 1 ρ 1) ) 1 ρ 1+ρ EL 2λ 1) ) ) 1 ρ 2] E L 1) ) ) 1 ρ + λ ρ Eθ j )EL 1) ) + EL 1) L 1) j )EB j ) ] 2λ EL 1) ) 1 ρ 2] E L 1) ) + EL 1) 1 ρ ) 2 ) ] λ 2 V arb ) + ρ 1 ρ 2λ EL 1) ) 1 ρ Summng the frst and thrd terms of the above equaton, we have EW ) = and E + ρ E L 1) L 1) ) 2] + EL 1) )1 ρ ) ρ 1 ρ )λ 2 V arb ) + ρ 2 ] 2λ 1 ρ )EL 1) ) ) 2] + λ 1 Eθ j )EL 1) ) + EL 1) L 1) j )EB j ) ] 2λ EL 1) ) 90) 91) EW down) s derved smlarly by usng Eq.6) wth T down) and X down): X down) = L 2) + A T down) = L 2) + A M L 2) ) ) j=+1 Y 2) j = L 2) 32 + j=+1 A θ down j 1 ) + A M j L 2) j )) )

33 From the dscusson above, when θ = θ down, the number of customers present at the varous channels at the begnnng of an cycle or a down cycle are probablstcally the same. Thus, denotng by L, the number of customers present at channel at the begnnng of cycle Cg, EL 1) ) = EL 2) ) = EL ) and E L 1) ) 2] = E L 2) ) 2] = E L ) 2]. Therefore, the total watng tme s gven by substtutng EW ) and EW down) n Eq. 87), wth L replacng of L 1) and L 2). Spressng the stars from all L and Cg, we wrte EW ) = ρ EL 2 ) + EL )1 ρ ) ρ 1 ρ )λ 2 V arb ) + ρ 2 ] 2λ 1 ρ )EL ) + EL2 ) + λ 1 Eθ j L ) + EL j L )EB j ) ] 4λ EL ) + EL2 ) + λ Nj=+1 Eθ down j 1 L ) + EL j L )EB j ) ] 4λ EL ) As E L we have, λ E,j 1 EB )L L j + θ j L + j=+1 θ down j 1 L EW ) = ρ EL 2 ) + EL )1 ρ ) ρ 1 ρ )λ 2 V arb ) + ρ 2 ] 2λ 1 ρ )EL ) = EL2 ), 92) + EL2 ) 2λ EL ) 93) To complete the evaluaton of EW ) as gven by Eq.93) ), t s just left to obtan EL 2 ). 6.3 Generatng Functons Let L 1) 1, L 1) 2,..., L 1) N ) be the vector state of the system at the begnnng of an cycle, and let L 2) 1, L 2) 2,..., L 2) N ) be the system s state at the begnnng of the followng down cycle or vca verca). Then, ] E z L2) 1 1, z L2) 2 2,..., z L2) N 1, L 1) 2,..., L 1) N L1) 33 N

34 = E exp λ j 1 z j ) R 1) k + θ k=1k j = θ N N λ j 1 z j ) R 1) k λ j1 z j )) 94) k=1k j where R m) k, the total net servce tme durng a vst of channel k, s R m) Lk =1 B k, m = 1, 2). Therefore, R k ω L k ) = Bk ω) ] L k, and E ] 1, z L2) 2 2,..., z L2) N N L 1) 1,..., L 1) N = θ N N λ j 1 z j ) k=1k j z L2) 1 However, k = Bk λ j 1 z j )) ] L 1) k. N N k=1k j Bk λ j 1 z j )) ] L 1) k = L N 1) 1 N = B 1 λ j 1 z j )) B 2 λ j 1 z j )) j 1 j 2 L 1) 2... N j N B N λ j 1 z j )) Settng, as before, L = L 1) = L 2), the generatng functon of the system s state at the begnnng of a cycle s gven by N E z L j j = G L z) = θ λ j 1 z j ) N E k=1 N j k L k B k λ j 1 z j )) = θ λ j 1 z j ) G L δ 1 z),..., δ N z)) 95) where δ k z) = N,j k Bk λ j 1 z j )). Now, takng dervatves, we obtan agan the set of equatons 80), leadng to Eq. 82): EL ) = λ 1 ρ )EC g ). That s, the total number of customers found by the server at channel at the begnnng of a cycle s equal to the number of customers who have arrved at that channel whle the server was away. Settng γz) = N λ j 1 z j ), and recursvely defne δ 0) = z, δ 1) z) = δ 1 z), δ 2 z),..., δ N z)), δ m) z) = L 1) N. 34

35 δ δ m 1) z) ) m = 2, 3,..., then, by teratng Eq. 95), we fnd, for every M = 1, 2,... G L z) = M 1 m=0 θ γ δ m) z) )) G L δ M) z) ) 96) It can be shown, smlarly to the calculaton n Boxma, Levy & Yechal 1992], that lm M δ M) z) = 1 and that the nfnte product ] m=0 E e γδm) z))θ converges f ρ < 1. Hence, G L z) = m=0 θ γ δ m) z) )) 97) The moments of the L k s can be now calculated n the regular manner. 7 Elevator-Pollng, Globally-Gated Servce Regme The cyclc-pollng, Globally-Gated GG) servce regme was ntroduced and studed by Boxma, Levy & Yechal 1992]. Under the cyclc GG procedure, at the start of each cycle, all customers present n the varous queues are marked, and those customers are the only ones to be served durng ths cycle. Customers arrvng n the mddle of a cycle wll be marked at the begnnng of the next cycle, and wll be served durng that cycle. Boxma, Levy & Yechal derved the LST of the cycle tme, C, and obtaned explct formulae for EC) and EC 2 ). Not surprsngly, t was shown, once more, that EC) = N=1 Eθ ) ] /1 ρ), where θ s the swtch-over tme from channel to channel + 1. They also derved the LST of W cyclc), the watng tme at queue, and obtaned an explct formula for E W cyclc)), namely, 1 E W cyclc)) = ρ j + ρ EC2 1 ) 2EC) + θ j 98) Furthermore, usng Eq. 98), t was shown that E W 1 cyclc)) < E W 2 cyclc)) <... < E W N cyclc)). Altman, Khamsy &Yechal 1992] then studed the Elevator-pollng wth GG servce regme. If θ = θ down, then, as n the Globally-Quas-Exhaustve 35

36 case, all and down cycles are probablstcly the same. Utlzng ths fact and result 98) they showed that, for the Elevator-GG model, mean watng tmes at all channels are the same, and equal to EW ) = 1 + ρ) EC2 ) 2EC) + Eθ) 2 99) Ths s the only known nonsymetrc pollng scheme that acheves such a farness phenomenon. 7.1 Cycle Tmes and Generatng functons In ths secton we extend the analyss gven by Altman, Khamsy & Yechal 1992]. We wll use the notaton of prevous sectons. The LSTs of the cycle tmes and the Generatng Functons of L 1) and L 2) are related to each other as follows. so that C 1 ω) = θ ω)g L 1) N G L 1)z) = E C2 E E { e ωc 1 L 1)} = θ ω) Z L1) N Ṽ L1) j j ω) Ṽ1 ω), Ṽ2ω),..., ṼNω) ). Also, C 2 = E C2 exp N λ j 1 z j )C 2 = C 2 λ j 1 z j ). Smlarly, G L 2)z) = C 1 N λ j 1 z j ) ). Thus, C 1 ω) = θ ω) C 2 λ j 1 Ṽ j ω) ), 100) and Hence, C 2 ω) = θ down ω) C 1 λ j 1 Ṽ j ω) ). 101) C 1 ω) = θ ω) θ down δω)) C 1 δ 2) ω) ) 102) 36

37 where, δω) = N λ j 1 Ṽ j ω) ), and δ 0) ω) = ω; δ m) ω) = δ δ m 1) ω) ), m = 1, 2, 3,... Iteratng, and notcng that lm m δ m) ω) = 0, so that lm m C1 δ m) ω) ) = 1 see Boxma, Levy & Yechal 1992]), we fnally have C 1 ω) = θ δ 2m) ω) ) θdown δ 2m+1) ω) ) 103) Smlarly, C 2 ω) = m=0 m=0 θ down δ 2m) ω) ) θ δ 2m+1) ω) ) 104) Now, f θ = θ down, then all and down cycles are the same, and equal to C 1. From Eq. 103) and 104), Hence, EC 1 ) = Eθ ) + ρec 2 ) and EC 2 ) = Eθ down ) + ρec 1 ). EC 1 ) = Eθ ) + ρeθ down ) 1 ρ 2 and EC 2 ) = Eθdown ) + ρeθ ) 1 ρ 2 105) Clearly, It readly follows now that EC) = EC 1 ) + EC 2 ) = Eθ ) + Eθ down ) 1 ρ 106) EL 1) ) = λ EC 2 ), EL 2) ) = λ EC 1 ) 107) and EY 1) ) = EL 1) )EV ) + Eθ ) = ρ EC 2 ) + Eθ ) 108) EY 2) ) = ρ EC 1 ) + Eθ down ) 109) 7.2 Mean Watng Tmes We can use agan Eqs. 6) and 31) to evaluate EW ). In order to calculate EW ) consder an arrval K to channel durng an cycle. K wll be served only durng the followng down cycle. Hence, the number of customers, served at queue durng the server s vst n whch K s beng served, s 37

38 T = A C 1 ), whle the number of customers present when the server enters channel s X Nj=+1 = T + A Y 2) ) j. Thus, as ET ) = λ EC 2 ) and ET 2 ) = λ 2 EC1)+λ 2 EC 1 ), the frst term n 6) when calculatng EW ) ) s gven by EC2 1 ) 2EC ρ). The probablstc nterrton of ths term wll ) become apparent n the sequel). Now, the numerator of the second term of 6) s gven by E Nj=+1 A C 1 )A Y 2) )) 2) j. Observe that Y j and C 1 are dependent snce Y 2) j = A j C 1 ) V jk + θj 1 down. We can complete the dervaton n ths manner, calculatng smlarly EW down), but nstead we choose to use a drect approach, applyng arguments smlar to those n Boxma, Levy & Yechal 1992]. Consder agan our customer K. Hs watng tme s composed of: ) The resdual part of the cycle, C R 1. ) The servce tme of all customers who arrved at queues +1, +2,..., N durng the cycle n whch K arrves. ) The swtch over tmes of the server on ts way down from channel N to channel. v) The servce tme of all customers who have arrved at channel durng the past part, C P 1, of the cycle n whch K arrves. Thus, EW ) = EC R 1 ) + j=+1 ρ j EC P 1 ) + EC R 2 ) ] + N 1 j= Eθ down j ) + ρ EC P 1 ) 110) Snce EC1 P ) = EC1 R ) = EC2 1 ) 2EC 1, we have see also Altman, Khamsy & ) Yechal 1992]), N 1 EW ) = ρ j + ρ EC1 R ) + Eθj down ) 111) j=+1 j= Smlarly, 1 EW down) = ρ j + ρ EC2 R 1 ) + Eθ j ) 112) 38

39 EW ) s readly obtaned by substtutng results 111), 112), 105) and 106) n Eq. 31). However, n order to complete the calculaton we need to evaluate EC 2 1) and EC 2 2). By dfferentatng 100) and 101) we get EC 2 1) = E θ ) 2) +2ρEθ )EC 2 )+ρ 2 EC 2 2)+EC 2 ) EC 2 2) = E θ down ) 2) + 2ρEθ down )EC 1 ) + ρ 2 EC 2 1) + EC 1 ) λ j EV 2 j ) 113) λ j EV 2 j ) 114) EC1) 2 and EC2) 2 are easly derved from 113) and 114) so that EC1 R ) and EC2 R ) are readly calculated. Now, EW ) s completely determned. In the case where θ = θ = θ down, then C 1 = C 2, so that EC 1 ) = EC 2 ) = Eθ) 1 ρ) EC1) 2 = EC2) 2 = 1 Eθ 2 ) + 2ρEθ)]2 + 1 ρ 2 1 ρ) Then, wth EC R 1 ) = EC R 2 ) we obtan N λ j EVj 2 ) ) Eθ) 1 ρ) 115) EW θ = θ down ) = 1 + ρ)ec R 1 ) + Eθ) 2 That s, all mean watng tmes are equal. 116) References 1] E. Altman, A. Khamsy and U. Yechal 1992), On Elevator Pollng wth Globally-Gated Regme, Queueng Systems 11, ] J.E. Baker and I. Rubn 1987), Pollng wth a General-Servce Order Table, IEEE Trans, on Communcatons 35, ] O.J. Boxma, H. Levy and U. Yechal 1992), Reservaton Schemes for Effcent Operaton of Multple-Queue Sngle-Server Systems, Annals of Operatons Research 35,

40 4] O.J. Boxma, J.A. Weststrate and U. Yechal 1993), A Globally Gated Pollng System wth Server Interrtons and Applcatons to the Reparman Problem, Prob. n Engneerng and Informatonal Scence, forthcomng. 5] S. Browne and U. Yechal 1989), Dynamc Prorty Rules for Cyclc- Type Queues, Adv. Appl. Prob. 21, ] S. Browne and U. Yechal 1991), Dynamc Schedulng n Sngle-Server Multclass Servce Systems wth Unt Buffers, Naval Research Logstcs 38, ] B.D. Bundary, J. Sztrk and R.B. Tapsr 1992), A Heterogeneous Scan Servce Pollng Model wth Sngle-Message Buffer, n Performance of Dstrbuted Systems and Integrated Communcaton, T. Hasegawa, H. Takag & Y. Tagahash, Eds.), North Holland, ] E.G. Coffman and M. Hofr 1982), On The Expected Performance of Scannng Dsks, Sam J. Comput. 11, ] Z. Moskovtch 1993), Global-Type Polces n Pollng Systems, n preparaton. 10] G.B. Swartz 1982), Analyss of a Scan Polcy n a Gated Loop System, n Appled Probablty-Computer Scence: The Interface, Vol 2, R.L. Dsney & T.J. Ott, Eds.), Brkhauser, ] H. Takag 1986),Analyss of Pollng Systems, MIT Press. 12] H. Takag and M. Murata 1986), Queueng Analyss of Scan-Type Pollng Systems, Computer Networkng and Performance Evaluaton, T. Hasegawa, H. Takag & Y. Takahash, Eds.), North Holland, ] H. Takag 1990), Queueng Analyss of Pollng Models: An date, n Stochastc Analyss of Computer and Communcaton Systems H. Takag, Ed.), North-Holland, ] K.S. Watson 1984), Performance Evaluaton of Cyclc Servce Strateges - a Survey, n Performance 84, E. Gelenbe, Ed.), North- Holland,