L. Donatiello & R. Nelson, Eds., Performance Evaluation of Computer. and Communication Systems, Springer-Verlag, 1993, pp

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1 L. Donatello & R. Nelson, Eds., Performance Evaluaton of Computer and Communcaton Systems, Sprnger-Verlag, 1993, pp ANALYSIS AND CONTROL OF POLLING SYSTEMS Ur Yechal Department of Statstcs & Operatons Research, School of Mathematcal Scences, Sackler Faculty of Exact Scences, Tel Avv Unversty, Tel Avv 69978, Israel Emal: Abstract. We present methods for analyzng contnuous-tme multchannel queueng systems wth Gated, Exhaustve, or Globally-Gated servce regmes, and wth Cyclc, Hamltonan or Elevator-type pollng mechansms. We dscuss ssues of dynamcally controllng the server's order of vsts to the channels, and derve easly mplementable ndex-type rules that optmze system's performance. Future drectons of research are ndcated. Keywords: Mult-channel queueng systems, pollng, gated, exhaustve, globally-gated, conservaton laws, Hamltonan tours, Elevator pollng, dynamc control. 1 Introducton Queueng systems consstng of N queues (channels) served by a sngle server whch ncurs swtch-over perods when movng from one channel 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 computer networks, telecommuncaton systems, multple access protocols, multplexng schemes n ISDNs, reader-head's movements n a computer's hard dsk, exble manufacturng systems, road trac control, repar problems and the lke. Very often such applcatons (e.g. Token Rng networks n whch N statons attempt to transmt ther messages by sharng a sngle transmsson lne) are modeled as a pollng system where the server vsts the channels n a cyclc routne or accordng to an arbtrary pollng table. In many of these applcatons, as well as n most pollng models, t s customary to control the amount of servce gven to each queue durng the server's vst. Common servce polces are the Exhaustve, Gated and Lmted regmes. Under the Exhaustve regme, at each vst the server attends the queue untl t becomes completely empty, and only then s the server allowed to move on. Under the Gated regme, all (and only) customers (packets, obs) present when the server starts vstng (polls) the queue are served durng the vst, whle customers arrvng when the queue s attended wll be served durng the next vst. Under the K -Lmted servce dscplne only a lmted number of obs (at most K ) are served at each server's vst to queue. There s extensve lterature on 630

2 the theory and applcatons of these models. Among the rst works are Cooper & Murray [1969] and Cooper [1970] who studed the cyclc Exhaustve and Gated regmes wth no swtchover tmes. Esenberg [1972] generalzed the results of Cooper & Murray by allowng changeover tmes and by consderng a general pollng table,.e., by allowng a general conguraton of the server's (perodc) sequence of vsts to the channels. Many other authors have nvestgated varous aspects of pollng systems, and for a more detaled descrpton the reader s referred to a book [1986] and an update [1990] by Takag, and to a survey by Levy & Sd [1990]. Recently, Globally-Gated regmes were proposed by Boxma, Levy & Yechal [1992], who provded a thorough analyss of the cyclc Globally-Gated (GG) scheme. Under the Globally-Gated regme the server uses the nstant of cycle begnnng as a reference pont of tme, and serves n each queue only those obs that were present there at the cycle-begnnng. A specal, yet mportant, pollng mechansm s the so-called Elevator (or scan)-type (cf. Shoham & Yechal [1992], Altman, Khamsy & Yechal [1992]): nstead of movng cyclcally through the channels, the server rst vsts the queues n one drecton,.e. n the order 1; 2; : : :; N (`up' cycle) and then reverses ts orentaton and serves the channels n the opposte drecton (`down' cycle). Then t changes drecton agan, and keeps movng n ths manner back and forth. Ths type of servce regme 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) derent tracks. Among ts advantages s that t saves the return walkng tme from channel N to channel 1. All the above models studed open systems wth external arrvals, where obs ext the system after servce completon. Altman & Yechal [1992] studed a closed system n whch the number of obs s xed. They analyzed the Gated, Exhaustve, Mxed and Globally-Gated regmes and derved measures for system's performance. One of the man tools used n the analyss of pollng systems s the dervaton of a set of mult-dmensonal Probablty Generatng Functons (PGS 's) of the number of obs present n the varous channels at a pollng nstant to queue ( = 1; 2; : : :; N). The common method s to derve PGF +1 n terms of PGF and from the set of N (mplct) dependent equatons n the unknown PGF 's one can obtan expressons whch allow for numercal calculaton of the mean queue sze or mean watng tme at each queue. The Globally-Gated regme stands out among the varous dscplnes as t yelds a closed-form analyss and leads to explct expressons for performance measures, such as mean and second moment of watng tme at each queue, as well as the Laplace-Steltes Transform (LST) of the cycle duraton. Most of the work on pollng systems has been concentrated on obtanng equlbrum mean-value or approxmate results for the varous servce dscplnes. Browne & Yechal [1989a], [1989b] were the rst to obtan dynamc control polces for systems under the Exhaustve, Gated or Mxed servce regmes. At the begnnng of each cycle the server decdes on a new Hamltonan tour and 631

3 vsts the channels accordngly. Browne & Yechal showed that f the obectve s to mnmze (or maxmze) cycle-duraton, then an ndex-type rule apples. Such a rule makes t extremely easy for practcal mplementatons. For the Globally- Gated regme Boxma, Levy & Yechal [1992] showed that mnmzng weghted watng costs for each cycle ndvdually, mnmzes the long-run average weghted watng costs of all customers n the system. A surprsng result holds for the Globally-Gated Elevator-type mechansm (Altman, Khamsy & Yechal [1992]): mean watng tmes n all channels are the same. In ths tutoral we present and dscuss () analytcal technques used n studyng pollng systems, and () methods derved and appled for dynamc control of such systems. In sectons 3 and 4 we present the basc tools for analyzng pollng systems wth Gated or wth Exhaustve servce regmes, respectvely. Secton 5 dscusses conservton laws and optmal vst frequences. In secton 6 we address the ssue of dynamc control of pollng systems havng servce regmes wth lnear growth of work. Secon 7 studes the Globally-Gated regme, and n secton 8 the Elevatortype pollng mechansm s analyzed. Future drectons of research are ndcated n secton 9. 2 Models and Notaton A pollng system s composed of N channels (queues), labeled 1; 2; : : :; N, where `customers' (messages, obs) arrve at channel accordng to some arrval process, usually taken as an ndependent Posson process wth rate. There s a sngle server n the system whch moves from channel to channel followng a prescrbed order (`pollng table'), most-commonly cyclc,.e., vstng the queues n the order 1; 2; ; : : :; N? 1; N; 1; 2; : : :. The server stays at a channel for a length of tme determned by the servce dscplne and then moves on to the next channel. Each ob n channel ( = 1; 2; : : :; N) carres an ndependent random servce requrement B, havng dstrbuton functon G (), Laplace-Steltes Transform eb (), mean b, and second moment b (2). The queue dscplne determnes how many obs are to be served n each channel. The dscplnes most often studed are the Exhaustve, Gated and Lmted servce regmes. To llustrate these regmes, assume the server arrves to channel to nd m obs (customers) watng. Under the Exhaustve regme, the server must servce channel untl t s empty before t s allowed to move on. Ths amount of tme s dstrbuted as the sum of m ordnary busy perods n an M=G =1 queue. Under the Gated regme, the server `gates o' those m customers and serves only them before movng on to the next channel. As such, the total servce tme n channel s dstrbuted as the sum of m ordnary servce requrements. Under Lmted servce regmes, the server must serve ether 1 ob, at most K obs, or deplete the queue at channel by 1 (.e., stay one busy perod of M=G =1 type). Accordng to the recently ntroduces Globally-Gated servce regme, at the start of the cycle all channels are `gated o' smultaneously, and only customers gated at that nstant wll be served durng the comng cycle. 632

4 Typcally, the server takes a (random) non-neglgble amount of tme to swtch between channels. Ths tme s called `walkng' or `swtchover' perod. The swtchover duraton from channel to the next s denoted by D, wth LST ed (), mean d, and second moment d (2). In some applcatons (e.g. star con- guraton) the tme to move from channel to channel ( 6= ) s composed of a swtch over tme D, out of channel, plus a swtch-n perod to channel, R. In other applcatons, even for a cyclc pollng procedure, the swtch-n tme R s ncurred only f there s at least one message n queue (see Altman, Blanc, Khamsy & Yechal [1992]), thus savng the swtchng tme nto an empty channel. We wll dscuss here only systems where each channel has an nnte buer capacty, assumng steady state condtons, and we focus on contnuous-tme models where channel s an M=G =1 queue wth Posson arrval rate and servce requrements B. The analyss wll concentrate on three man servce regmes: Gated, Exhaustve and Globally-Gated. 3 Analyss of the Gated Regme Let denote the number of obs present n channel ( = 1; 2; : : :; N) when the server arrves at (polls) channel (= 1; 2; : : :; N). = ( 1; 2 ; : : :; N ) s the state of the system at that nstant. Let A (T ) be the number of Posson arrvals to channel durng a (random) tme nterval of length T. Then, for the Gated servce regme, the evoluton of the state of the system s gven by 8 > < = +1 >: + A P k=1 B k + D ; 6= A P k=1 B k + D ; = where B k are all dstrbuted as B. One of the basc tools of analyss s to derve the multdmensonal Probablty Generatng Functon (PGF ) of the state of the system at the pollng nstant to channel ( = 1; 2; : : :; N). PGF s dened as G (z) = G (z 1 ; z 2 ; : : :; z?1 ; z ; z +1 ; : : :; z N ) = E Then, for the Gated regme, whle usng (1), G +1 (z) = E NY 2 = E 6 4 N Y z +1 6= NY NY P z E z A ( k=1 3 NY Bk) 7 5 E 633 z z A (D) (1) : (2) (3)

5 For a Posson random varable A (T ), and wth e T () denotng the Laplace- Steltes Transform (LST) of T, we have and Therefore G +1 (z) = E 2 6 E[z A(T ) ] = E T [e?(1?z)t ] = T e (1? z ) 4 N Y NY E 6= z z A (T ) eb N = T e (1? z ) : (1? z ) 3 7 N 5 D e Thus, for = 1; 2; : : :; N? 1; N (where we take N + 1 as 1) 1 ; z +1 ; : : :; z N G +1 (z)=g z1 ; z 2 ; : : :; z?1 ; e B (1?z ) (1? z ) : A D e (1?z ) (4) Equatons (4) dene a set of N relatons between the varous PGFs whch are used to derve moments of the varables, as follows. Moments The mean number of messages, f () = E( ), present n channel at a pollng nstant to channel s obtaned by takng dervatves of the PGFs, where f () = E( ) (z) z=1 A set N 2 lnear equatons n f () : ; = 1; 2; : : :; N determnes ther values: f () + f +1 () = b f () + d b f () + d 6= = (6) Indeed, equatons (6) could be obtaned drectly from (1). by P P N Set k = k b k, = N k=1 k, d = 8 < P h?1 k= d k 1? + d k f () = : d 1? k=1 d k. Then, the soluton of (6) s gven 6= = The explanaton of (7) s the followng. It wll be shown shortly that the mean cycle tme s E[C] = d=(1? ). Durng that tme the mean number of arrvals to channel s E[C]. Also, durng a cycle the server renders servce to channel k for an average length of tme k E[C]. Thus, the elapsed tme snce the last gatng nstant of channel ( 6= ) untl the pollng nstant of channel, s 634 (7)

6 P?1 k= k E[C] + d k. Wthn that tme-nterval the mean number of arrvals to channel s f (), as gven by (7). The second moments of the are also derved from the set of PGFs (4). Let f (; k) = E[ k ] k z=1 f (; ) = E (? 1) ] G 2 (; ; k = 1; 2; : : :; N not all equal) z=1 (8)? Clearly, Var[ ] = f (; ) + f ()? f () 2. Takng dervatves, the soluton of (8) s gven (see, Takag [1986]) as a set of N 3 lnear equatons n the N 3 unknowns f (; k). Cycle Tme The mean cycle tme s obtaned from the balance equaton E[C] = E[C] + d. Hence, E[C] = d 1? : The mean soourn tme of the server at channel s f ()b = E[C], and the number of obs served n a cycle s clearly, P N =1 f () =?P N =1 E[C]. The PGF of L and Watng Tmes Consder the probablty generatng functon, Q (z) = E(z L ), of the number of customers, L, left behnd by an arbtrary departng customer from channel n a pollng system wth arbtrary servce regme. As the dstrbutons of the 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 condton at an arbtrary pont of tme. 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 t. Then the PGF of L s gven by (see, Takag [1986], p. 78) Q (z) = E?P T n=1 zl(n) E(T ) : (9) As L (n) =? n + A?P n k=1 B k, the evaluaton of the expresson for Q (z) becomes Q (z)= 1 E(T ) E T n=1 z?n+a( P n k=1 Bk) = 1 E(T ) E z T n=1 z?n+a( P n k=1 Bk) 635

7 = 1 E(T ) E z = T n=1 z?n e?( P n k=1 Bk)(1?z) = 1 0 B = 1 E(T ) eb? (1? z) z E(T ) E 1? T z h eb((1?z)) z n=1 T 1? eb((1?z)) z eb? (1?z) n! 1 A C? eb (1? z) E(T ) z? B e? E z?t? z T? eb T ( (1? z)) : (10) (1? z) Let W q denote the queueng tme of an arbtrary message at queue, and let W = W q + B denote the soourn (resdence) tme of a message n the system. As the messages left behnd by a departng message from channel have all arrved durng ts resdence tme W, we have Q (z) = Hence, 1 P number of messages at channel = k z k = 1 z k Z 1 k=0 k=0 0 = W f (1? z) = Wq f (1? z) B e (1? z) fw q (s) = Q (1? s= ) eb (s) For the Gated regme, = T, and therefore Q (z) = eb? (1? z) e?w ( w) k dp (W w) k! E(T ) z? e B ( (1? z)) E[z ]? E ( e B ( (1? z))) (see also Takag [1986], p. 109). As E(T ) = E( ) = E[C], usng (11) and (12) leads to z (11) (12) E? W q = E? ( )2? E( ) 2 E( ) (1 + ) = (1 + )f (; ) 2 2 E[C] (13) By Lttle's law, E[L ] = E(Wq ) + b. 4 Exhaustve Regme To derve the PGF of the state of the system at a pollng nstant to channel + 1 we use the law of moton +1 = 8 < : + A A (D ) ; P k=1 k + D ; 6= = (14) where denotes the length of a regular busy perod n an M/G /1 queue, and k are all dstrbuted as. Then, 636

8 Hence, NY G +1 (z) = E 2 = E 6 = E N Y z +1 6= 4 N Y 6= z z NY E 6= e 6= z A ( P k=1 k) (1? z ) 0 G +1 (z)=g 1 ; z 2 ; : : :; z?1 ; e (1?z ) 6= 3 7 NY 5 A (D) E z D e (1? z ) 1 ; z +1 ; : : :; z N C A D e (1?z ) (15) To get the N 2 values of f () one can derentate (15) or use drectly (14). The result s f () + f +1 () = E( )f () + d d 6= = (16) where E( ) = b =(1? ) s the mean duraton of a regular busy perod at channel. The soluton of (16) s f () = 8 >< >: P?1 d k=+1 k 1? (1? ) d 1? P?1 + k= d k 6= = The nterpretaton of (17) s the followng. The mean cycle tme s agan E[C] = d=(1? ), whch s derved from the same balance equaton as for the Gated regme. The fracton of tme that the server stays at channel s, hence, durng the tme nterval snce the server leaves (an empty) channel untl t arrves there agan, the mean number of accumulated messages at s (1? )E[C]. For channel 6=, the total swtchover tmes P from the moment?1 the server last exted the channel untl t enters channel s k= d k, and the mean tme spent n each of the channels k = + 1; + 2; ; : : :;? 1, s k E[C]. Thus, the expected number of?p obs accumulated P at channel when the server polls channel s gven by?1? k= d?1 k + d k=+1 k 1?. The PGF of the number of messages at channel can be obtaned by usng result (10). For the Exhaustve case, the number of customers served durng a vst to channel s T =?P + A k=1 k, so that E(T ) = f () + f ()E( ) = f ()=(1? ), and by usng (17), E(T ) = E[C]. 637 (17)

9 The PGF of the number of messages at channel at an arbtrary pont of tme s gven by Takag [1986], p. 79: Q (z) = 1 E[C] e B (1? z) z? e B (1? z) E[z ]? 1 (18) The mean number of messages at channel and the mean queueng tmes are derved from (18), E[L ] = + 2 b(2) 2(1? ) + f (; ) 2 (1? )E[C] E[W q ] = b (2) 2(1? ) + f (; ) 2 2 (1? )E[C] Agan, the values of f (; ) have to be calculated numercally by solvng a set of N 3 lnear equatons n the unknowns f (; k) derved (see (8)) by derentatng the PGFs n (15). Remarks on Computatonal Methods Several numercal procedures have been proposed for computng the mean watng tmes n pollng systems wth Gated or wth Exhaustve servce regmes. The procedure mentoned above of determnng the mean delay n varous channels by solvng a set of N 3 lnear equatons s called the Buer Occupancy method. It s of hgh computatonal complexty, but can also be appled to solve models wth swtch-n tmes or wth lmted-servce regmes. A more ecent procedure s known as the Staton Tme method (see Ferguson & Amnetzah [1985]). Ths s an teratve procedure whch has been appled to a number of pollng systems, but cannot be drectly used for closed networks or for open systems wth customers' routng. Sarkar & Zangwll [1989] have developed an algorthm for cyclc (Exhaustve or Gated) systems where the mean watng tmes are obtaned by solvng a set of only N lnear equatons (thus requrng O(N 3 ) computatonal steps). Recently, Konhen, Levy & Srnvasan [1993a] ntroduced a Descedant Set (DS) approach whch s based on countng the number of descedants generated n the system by each customer. The method can be appled to varatons of Exhaustve or Gated pollng systems whch are based on xed order of vsts, and can also be used to derve second and hgher delay moments. It s clamed that the DS s superor to other methods due to ts low computatonal complexty, even though t s based on the buer occupancy varables. In a further eort to develop ecent computatonal methods, the same authors [1993b] ntroduced the Indvdual Staton (IS) technque whch, lke the DS procedure, allows for the determnng of mean watng tme at one or more selected nodes wthout havng to obtan mean watng tmes at all channels smultaneously. The IS s superor to the DS for systems wth hgh utlzaton factor, whle the DS would be preferred for systems wth very large N. 638

10 5 Conservaton Laws and Vst Frequences In an arbtrary sngle-server system (wth sngle or multple queues) when no work s generated or lost wthn the system, the amount of work present does not depend on the order of servce { and hence equals the amount of work n the `correspondng' system wth a sngle queue and FCFS servce dscplne. Ths `prncple' of work conservaton yelds useful expresson whch we now dscuss. Suppose that no swtchng tmes are ncurred n our pollng system, and assume cyclc or any order of the server's vsts. Then t s well known (see, Klenrock [1975]) that the expected amount of work n the system s constant,.e., P N =1 b E[L ] = E(W ) = b (2) W : (19) 2(1? ) =1 =1 When swtchng tmes are ncurred, Boxma & Groenendk [1987] and Boxma [1989] have derved the so called `pseudo-conservaton laws' and showed that for an arbtrary pollng system wth mxed channels =1 E(W ) = W + d(2) 2d + =1 d 2(1? ) E(W ) = W + d(2) 2d + 2? =1 d 2(1? ) 2 + =1 EM (1) (20) where EM (1) s the expected unnshed work at the th queue at an (arbtrary) nstant of departure of the server from that queue. Result (20) holds for any servce regme, and EM (1) depends only on the servce dscplne n channel. For the Exhaustve servce regme EM (1) = 0 for every, so that 2? : (21) For the Gated regme, we use (7) and wrte Hence, for the Gated, =1 EM (1) = (f ()b ) b = 2 E(W ) = W + d(2) 2d + d 1? =1 : 2 d : (22) 2(1? ) =1 It follows that for the same set of parameters, whenever swtchover tmes are ncurred the mean amount of work n the system under the Exhaustve regme s smaller then that under the Gated dscplne. Furthermore, expressons (21) and (22) enabled Boxma, Levy & Weststrate (see, Boxma [1991]) to develop `good' vst frequences of the server to the varous channels so as to construct a pollng table that wll reduce the value of the expected amount of work n the system, 639

11 as expressed n (20). For the Exhaustve and for the Gated regmes the vst frequences v exh, and v gated are gven by p (1 v exh? )=d = p (1? )=d PN v gated = PN p (1 + )=d p (1 + )=d For example, n a 3-channel case for whch the calculated vst frequences are 0.52, 0.32 and 0.16, the approxmate vst frequences are 1/2, 1/3 and 1/6, respectvely, such that a (perodc) pollng table of sze 6 s constructed wth the order of vsts [1,2,1,3,1,2]. Another approach n the attempt to control and otmze the vst frequences of the server to the varous channels s the Cyclc Bernoull Pollng (CBP) ntroduced by Altman & Yechal [1993]. The server moves cyclcally among the N channels where change-over tmes between statons are composed of two parts: walkng tmes requred to `move' from one channel to another and swtch-n tmes that are ncurred only when the server actually enters a staton to render servce. Upon arrval to channel the server swtches n wth probablty p, or moves on to the next channel (wth probablty 1? p ) wthout servng any customer. Altman & Yechal analyzed the Gated and Exhaustve regmes and dened a mathematcal program to nd the optmal values of the swtch-n probabltes fp g N =1 so as to mnmze the expected amount of unnshed work n the system. Any CBP scheme for whch the optmal p 's are not equal to 1 yelds a smaller amount of expected unnshed work n the system than that n the standard cyclc procedure wth equvalent parameters. They showed that even n the case of a sngle queue, t s not always true that p 1 = 1 s the best strategy, and derved condtons under whch t s optmal to have p 1 < 1. 6 Dynamc Control of Server's Vsts: Hamltonan Tours A basc queston that arses when plannng ecent pollng systems concerns the order of vsts performed by the server. For statc order one can thnk of a `good' pollng table that optmzes some measure of eectveness. Steps n ths drecton were taken, as mentoned n secton 5, by varous authors. However, a more reachng goal s to control the system dynamcally, so that the server wll modfy ts order of vsts n response to the stochastc evoluton of the system. In other words, the general control problem facng the server when t exts a specc channel, s \whch of the channels to vst next?". In tryng to solve ths problem Browne & Yechal [1989a], [1989b] developed and formulated sem-markov Decson Processes (SMDP) for the Gated and for the Exhaustve regmes. They derved a set of optmalty equatons where the obectve s to mnmze mean weghted watng costs. However, these equatons are non-tractable, 640

12 so that one should look for alternatve methods. An appealng approach s to look for sem-dynamc control schemes. The dea s to dspatch the server to perform Hamltonan tours, each tour derent from ts prevous one, dependng on the state of the system at the begnnng of the tour, so as to optmze some measure of eectveness. Speccally, suppose that at the begnnng of a cycle the state of the system s (n 1 ; n 2 ; : : :; n N ), where n s the number of obs watng n channel (1 N). Assume for the moment that swtchng tmes between channels are neglgble. The obectve s to choose a path (Hamltonan tour) through the queues so as to mnmze the expected tme of traversng ths path. It was shown by Browne & Yechal [1989a], [1989b] that for both servce dscplnes { the fully Gated and the fully Exhaustve { ths measure of eectveness s mnmzed f the channels are ordered by ncreasng values of the ndex n =. Ths s a surprsng result, as the ndex n = does not nclude the servce tmes at the varous channels. It s surprsng as well that the same ndex-rule holds for both servce regmes (although, obvously, the duraton of a Gated-type cycle that starts wth (n 1 ; n 2 ; : : :; n N ) ders from ts Exhaustve counter-part startng wth the same system-state). The dynamcs of the control are such that at the end of each Hamltonan cycle a new system-state s observed, say (n 0 1; n 0 2; : : :; n 0 N ), and the server follows a new path governed by a new order: ncreasng values of n 0 =, etc. Ths s an extremely smple rule whch can be drectly mplemented. Moreover, suppose that, for one reason or another, there are systems where the obectve s to maxmze the duraton of each cycle. Then, the ndex-rule that determnes the order of vsts to the channels s smply reversed: the server completes a Hamltonan tour determned by a decreasng order of n =. To understand the above surprsng result Browne & Yechal [1990] studes a general schedulng problem wth a lnear growth of work, as follows. Consder a sngle-processor system wth N obs watng to be performed sequentally. Let a be the ntal (expected) processng tme requrement of ob ( = 1; 2; : : :; N), called the `core'. If ob s delayed and s started at tme t, then ts processng requrement grows lnearly wth the delay to Y (t) = a + t where s the growth rate of work requrement by ob. Consder the processng order 0 = (1; 2; : : :; P N), and let Y denote the actual processng length of ob k under 0. Let S k = Y =1 be the completon tme of ob k under 0 (S 0 = 0). Then Y = a + S?1. By addng S?1 to both sdes we obtan a set of derence equatons S? (1 + )S?1 = a ( = 1; 2; : : :; N) (23) The soluton of (23) s S = =1 a Y r=+1 (1 + r ) ( = 1; 2; : : :; N) (24) 641

13 so that the makespan s S N = S N ( 0 ) = P N =1 a Q N r=+1 (1 + r). The obectve s to nd a vst order that mnmzes the makespan S N () over all n! possble permutatons. Consder now the processng sequence 1 = (1; 2; : : :;? 1; + 1; ; + 2; : : :; N), where the order of obs and + 1 s nterchanged. The correspondng makespan s S N ( 1 ). Then, t s easy to show that S N ( 0 ) < S N ( 1 ) a = < a +1 = +1. That s, the makespan s mnmzed (maxmzed) f we process the obs n an ncreasng (decreasng) order of the rato ndex a =,.e., `core' dvded by `growth rate'. Consder agan the Gated regme. If (n 1 ; n 2 ; : : :; n N ) s the state of the system at the start of the Hamltonan tour, then a = n b. The growth rate (.e., the amount of work owng to channel per unt of tme) s. Hence, a = n b b = n :? For the Exhaustve regme, a = n E( ) = n b 1?, whereas = 1? (the duraton of tme that the server has to stay n channel grows lnearly at a rate b of 1? for each new arrval. As the rate of arrvals s, we have = 1? ). Thus, for the Exhaustve case a = n b 1? = n b 1? = n whch s the same ndex as for the Gated regme. We can now rentroduce the swtchover and swtch-n tmes. For llustraton, assume a star-conguraton of the system. Recall that D s the swtchover tme out of and R denotes the swtch-n duraton nto. Then, for the Gated regme, assumng gatng occurs after swtch-n s completed, a = n b + (1 + )r + d = ; so that a = = n b + (1 + )r + d. For the Exhaustve r a = + n b + d 1? 1? = =(1? ) ; so that a = = r + n b + d (1? ). It should be emphaszed that the schedulng prncple a = can be appled to any system wth a mxed set of servce regmes among the channels: Gated, Exhaustve, Bnomal or Bernoull Gated, Bnomal or Bernoull Exhaustve, etc. (see, Yechal [1991]). All that one has to do s to calculate (once) for every channel, and then, at the begnnng of each new Hamltonan tour, to calculate the current `core' a at each channel. Then, performng a vst tour that follows an ncreasng (decreasng) order of a = wll mnmze (maxmze) cycle duraton. 642

14 Browne & Yechal [1991] further employed the above deas to acheve dynamc schedulng n systems wth only a unt buer at each channel. 7 The Globally-Gated Regme A drawback both of the Gated and the Exhaustve regmes s that they are not `far' wth regard to the FCFS prncple. To help resolve ths dchotomy, Boxma, Levy & Yechal [1992] ntroduced a (cyclc) Globally Gated (GG) servce scheme whch uses a tme-stamp mechansm for ts operaton: the server moves cyclcally among the queues, and uses the nstant of cycle-begnnng as a reference pont of tme; when t reaches a queue t serves there all (and only) customers who were present at that queue at the cycle-begnnng. Ths strategy can be mplemented by markng all customers wth a tme-stamp denotng ther arrval tme. In ts nature the GG polcy resembles the regular Gated polcy. However, the GG polcy leads to a mathematcal model whch allows for dervaton of closed-form expressons for the mean delay n the varous queues. As a result, the operaton of the pollng system by the GG polcy s easy to control and optmze. As n earler sectons, the system conssts of N nnte-buer channels, the P rate of N oered load to queue s = b and the total system load-rate s =1. When leavng queue and before startng servce at the next queue, the server ncurs P a random swtchover perod D. The total `walkng' tme n a cycle s N D D =1. (Clearly, other `Global' versons, such as Globally Exhaustve, can be easly magned and analyzed.) Cycle Tme Assume, wthout loss of generalty, that a cycle starts from channel 1. Let ( 1 1 ; 2 1 ; : : :; 1 ; : : :; N 1 ) = ( 1; 2 ; : : :; ; : : :; N ) be the state of the system at the begnnng of the cycle. Then, the cycle duraton s C = D + The LST of C s derved as follows k=1 B k : E? e?wc ( 1 ; 2 ; : : :; N ) = e D(w) N Y? eb (w) : (25) On the other hand, the length of a cycle determnes the ont queue-length dstrbuton at the begnnng of the next cycle. Hence E NY z 2 NY = E C 4 E z C = e C N 3 5 = EC 2 4 exp? (1? z )C 3 5 (1? z ) : (26) 643

15 Combnng (25) and (26) ec(w) = e D(w) e C? 1? e B (w) : (27) The mean cycle tme s derved from (27) E[C] = d + b E[C] : That s, E[C] = d=(1? ), as for the Gated and the Exhaustve regmes. The second moment of C s derved from (27) 2 3 E[C 2 ] = 4 d (2) + 2d + b (2) E[C] 5 (1? 2 ) : (28) Let C P and C R denote, respectvely, the past and resdual duraton of a cycle. It s well known that and E[C P ] = E[C R ] = E[C2 ] 2E[C]. ec P (w) = e CR (w) = 1? e C(w) we[c] Pseudo-Conservaton law To derve a pseudo-conservaton law we use (20) and the observaton that for the cyclc GG regme, E( ) = E[C] and EM (1) " = =1 E( )b + Substtutng (29) n (20) yelds?1 # d =1 =?1 =1 E(W ) = W + d(2) 2d + d 1? 2 + d 1? + d =2?1 =1 d + 2 1? : (29) d : (30) Watng Tmes Consder an arbtrary ob K at channel k. The cycle age at the ob's arrval nstant s C P. The ob's watng tme s composed of () the resdual cycle tme C R, () the servce tmes of all customers who arrve at channels 1 to k? 1 durng the cycle n whch K arrves, () the swtchover tmes of the server through channels 1 to k, and (v) the servce tmes of all customers that arrve at channel k durng the past part of the cycle, C P. Then 644

16 k?1 E(W k ) = E[C R ] + = It readly follows that k?1? k?1 E[CP ] + E[C R ] + k?1 + k E[C R ] + E(W k+1 )? E(W k ) = ( k+1 + k )E[C R ] + d k so that, for the cyclc GG regme, we always have d + k E[C P ] d : (31) E(W 1 ) < E(W 2 ) < : : : < E(W N ) : (32) Boxma, Weststrate & Yechal [1993] extended the cyclc GG model to the case where the server suers perods of breakdown, and appled the results to realworld reparman problems where both preventve and correctve mantenance actons are consdered. Statc Optmzaton Let c k be the cost rate of a watng ob at queue k. Then, the mean weghted watng cost of an arbtrary ob n the system s N k k=1 c k E(W k ) : (33) By substtutng (31) nto (33) and usng an nterchange argument t follows that the cycle whch mnmzed (33) s determned by an ncreasng order of the ndex u = 2E[C R] + d c If d s neglgble, the above ndex reduces to the ndex b =c, whch s the well known \c" rule. Dynamc Control An mportant characterstc of the GG regme s that the order of vsts selected for one cycle does not aect the future stochastc behavour of the system. Moreover, any Hamltonan tour that starts from state (n 1 ; n 2 ; : : :; n M ) yelds the same cycle duraton C(n 1 ; n 2 ; : : :; n N ). Thus, f we consder the costs ncurred durng the cycle by the customers present at ts ntaton and add to t the costs ncurred along that cycle by the new arrvals, then the long-run mmnal cost can be acheved by determnng a new optmal Hamltonan tour for each cycle ndependently. The mean total weghted cost ncurred durng a cycle startng wth (n 1 ; n 2 ; : : :; n N ) s 645

17 k=1 c k n k k?1 (n b + d ) + b k n k?1 =1 (34) + c k k E C(n 1 ; n 2 ; : : :; n N ) 2 2 k=1 where the rst term s the contrbuton to total cost ncurred by the customers present at the cycle begnnng, and the second term s due to the customers arrvng durng that cycle (see, Yechal [1976]). The only term n (34) that depends on the order of vsts s P N k= c kn k P k?1 (n b + d ). It follows (by an nterchange argument) that the optmal order of vsts that mnmzes expected total costs of the comng cycle s determned by an ncreasng order of the (Gttns) ndex n b + d n c : Agan, for neglgble d ths ndex reduces to the \c" rule (.e., b =c ). 8 Elevator-Type Pollng ; 1 nup 2 N In an Elevator-type (scan) pollng mechansm the server alternates between `up' and `down' cycles. In an `up' cycle t vsts the channels n the order 1; 2; : : :; N? 1; N, and n a `down' cycle the order of vsts s reversed to N; N?1; : : :; 2; 1. Ths type of pollng procedure s encountered n many applcatons, e.g., t models a common scheme of addressng a hard dsk for wrtng (readng) nformaton on (from) derent tracks. It s mportant to note that the Elevator-type pollng saves the return walkng tme from channel N to channel 1. A comprehensve analyss of Elevator-type pollng wth four derent servce regmes can be found n Shoham & Yechal [1992]. Here we present the Globally-Gated (GG) regme as dscussed n Altman, Khamsy & Yechal [1992]. Accordng to the Elevator-type pollng wth GG servce regme all channels are gated o at the begnnng of the `up' cycle, where the system-state s (n up ; : : :; nup), and the server resdes n channel for nup regular servce duratons. At the end of the up cycle all channels are gated agan, the system-state s (n down 1 ; n down 2 ; : : :; n down N ), and the server starts ts down cycle, servng ndown customers n channel. We assume that the down walkng tme from channel + 1 to channel has the same dstrbuton as the up walkng tme D from channel to channel + 1. A key observaton s that arbtrary up and down cycles have the same dstrbuton, whch ders from ts cyclc GG counter-part only n that t s smaller by the `saved' walkng tme D N. Hence, the results derved for the cycle tme dstrbuton (27) and for mean watng tmes (31) n a cyclc GG regme are drectly applcable to the Elevator case, wth D N =

18 Watng Tmes Consder an arbtrary ob at channel k. Snce all cycles are dstrbuted alke, the ob arrves durng an up or a down cycle wth equal probabltes, 0.5. Hence, ts mean watng tme s gven by E(W k ) = 0:5E The expresson for E W k W k W k server moves up server moves down server moves down by reversng the order of vsts, we have E = server + 0:5E W k moves up : (35) s gven by (31), wth d N = 0, whereas, =k+1 N?1 + k E[C R ] + d : (36) Combnng (35) wth (31) and (36) yelds the surprsng result =k E(W k ) = (1 + )E[C R ] + 0:5d : (37) That s, expected watng tmes are equal n all channels. Ths s the only-known non-symmetrc pollng system that exhbts such a \farness" phenomenon. An explanaton of result (37) s the followng. An arbtrary arrval has to wat, on the average, E[C R ] unts of tme untl the cycle (up or down) n whch t arrves termnates. Then, t wats untl the server moves back to channel k, whch requres, on the average (takng nto account both drectons), 1 2? E[CR ]+ E[C p ] + d unts of tme. Optmal Arrangement of Channels The nterestng result that E(W k ) s the same for all channels, ndependent of ther locaton, leads to consderng channels' arrangement such that the varaton n watng tmes wll be small. Let a = 2E[C R ] + d ( = 1; 2; : : :; N). Then E W k server moves down server E W k moves up k?1 = E[C R ](1 + k ) + = E[C R ](1 + k ) + =1 =k+1 a a + d k P P Let k P = E(W k down)? E(W P k?1 k up) = a N =1? a =k+1? d k. Now, N 1 =? a N?1 =2? d 1 < 0, N = a =1 > 0 (recall that d N = 0), and k s a monotone ncreasng functon of k. One goal s to arrange the channels such that max 1kN k s as small as possble. Clearly 647

19 max k 1kN = max 1 ; N = max a? 2E[C R ] 1 ; =1 =1 a? 2E[C R ] N (38) It follows from (38) that max 1kN k s mnmzed f channel 1 s the one wth the hghest value of and channel N s the one wth the second hghest value of (or vce versa). 9 Future Drectons of Research We have presented methods of analyss for sngle-server, contnuous-tme, nnte buers pollng systems, and studed several control and optmzaton problems. Dcult problems are nte-capacty models and lmted servce regmes, for whch only partal solutons are gven n the lterature (see, bblography n Takag [1990]). A few authors have studed pollng systems wth multple servers, and recently Browns & Wess [1992] nvestgated dynamc prorty rules for a system wth parallel servers. All the systems mentoned above are open, wth external arrvals, where obs ext the system after servce completon. Closed systems should also be nvestgated, and only recently Altman & Yechal [1992] analyzed such systems wth Gated, Exhaustve or Globally-Gated servce regmes. For other future drectons of research we state a recent `call for papers' on \Dscrete-Tme Models and Analyss Methods": \The past few years have seen an ncreasng nterest n dscrete-tme models and ther soluton technques. One of the drvng forces behnd ths area has been new developments n telecommuncatons, espacally n hgh-speed metropoltan area and wde area networks. Tehcnologcal advances and user demands have shfted the evoluton of telecommuncaton systems towards ntegrated networks where nformaton s transferred n small, ofted xed-sze, packets, slots or cells (e.g., ATM networks, hghspeed LANs and MANs such as DQDB, etc...), opertng n a dscrete-tme envronment. The resultng mathematcal models of such slotted systems, crucal for the evaluaton of desgn alternatves and ther dmensonng, are dscrete-tme models. The complexty of the stochastc processes nvolved (e.g., arrval and departure processes) and of the system operaton mechansms (e.g., servce mechansm, access protocol, etc...) pose an exctng challenge for the development of ecent and tractable methods for dervng the man performance measures of these systems. Papers are solcted on dscrete-tme models and ther analyss methods, n partcular on, but not restrcted to, the followng topcs: Dscrete-tme queueng models (pollng systems, prorty systems, multserver systems, vacaton models, etc...). 648

20 Exact and approxmate soluton methods for dscrete-tme queueng models, wth emphass on the ecency and the numercal tractablty of these methods. Stochastc processes as trac models for performance studes (takng nto account the dversty of tme scales, correlatons between arrvals, etc...) Dscrete-tme markov chans and ther analyss methods". Naturally, we add to the above topcs the nterestng and challengng problems of control and optmzaton of such systems. Bblography 1. Altman, E., Blanc, H., Khamsy, A., Yechal, U.: Gated-type pollng systems wth walkng and swtch-n tmes. Techncal Report, Dept. of Statstcs & OR, Tel Avv Unversty Altman, E., Khamsy, A., Yechal, U.: On elevator pollng wth globally-gated regme. Queueng Systems 11 (1992) Altman, E., Yechal, U.: Pollng n a closed network. Techncal Report SOR-92-14, Dept. of Statstcs & OR, New York Unversty Altman, E., Yechal, U.: Cyclc Bernoull pollng. ZOR-Methods and Models of Operatons Research 38 (1993). 5. Boxma, O.J.: Workloads and watng tmes n sngle-server systems wth multple customer classes. Queueng Systems 5 (1989) Boxma, O.J.: Analyss and optmzaton of pollng systems. In: Cohen, J.W., Pack, C.D. (Eds.) Queueng, Performance and Control n ATM. North-Holland, 1991, pp Boxma, O.J., Groenendk, W.P.: Pseudo conservaton laws n cyclc servce systems. Journal of Appled Probablty 24 (1987) Boxma, O.J., Levy, H., Yechal, U.: Cyclc reservaton schemes for ecent operaton of multple-queue sngle-server Systems. Annals of Operatons Research 35 (1992) Boxma, O.J., Weststrate, J.A., Yechal, U.: A globally gated pollng system wth server nterruptons, and applcatons to the reparman problem. Probablty n the Engneerng and Informatonal Scences 7 (1993). 10. Browne, S., Yechal, U.: Dynamc prorty rules for cyclc-type queues, Advances n Appled Probablty 21 (1989a) Browne, S., Yechal, U.: Dynamc routng n pollng systems. In: M. Bonatt (Ed.) Teletrac Scence for New Cost-Eectve Systems, Networks and Servces. North- Holland, 1989b, pp Browne, S., Yechal, U.: Schedulng deteroratng obs on a sngle processor. Operatons Research 38 (1990) Browne, S., Yechal, U.: Dynamc schedulng n sngle-server multclass servce systems wth unt buers. Naval Research Logstcs 38 (1991) Browne, S., Wess, G.: Dynamc prorty rules when pollng wth multple parallel servers. Operatons Research Letters 12 (1992) Cooper, R.B. Murray, G.: Queues served n cyclc order. Bell System Techncal Journal 48 (1969)

21 16. Cooper, R.B.: Queues served n cyclc order: watng tmes. Bell System Techncal Journal 49 (1970) Esenberg, M.: Queues wth perodc servce and changeover tme. Operatons Research 20 (1972) Ferguson, M.J., Amnetzah, Y.J.: Exact results for nonsymmetrc token rng systems. IEEE Transactons on Communcatons 33 (1985) Klenrock, L.: Queueng Systems, Vol. 1: Theory. John Wley, Konhem, A.G., Levy, H., Srnvasan: Descendant set: an ecent approach for the analyss of pollng systems. IEEE Transactons on Communcatons (to appear 1993a). 21. Konhem, A.G., Levy, H., Srnvasan: The ndvdual staton technque for the analyss of pollng systems. Techncal Report, 1993b. 22. Levy, H., Sd, M.: Pollng systems: applcatons, modelng and optmzaton. IEEE Transactons on Communcatons 8 (1990) Sarkar, D., Zangwll, W.I.: Expected watng tme for nonsymmetrc cyclc queueng systems { exact results and applcatons. Management Scence 35 (1989) Shoham, R., Yechal, U.: Elevator-type pollng systems. Techncal Report, Dept. of Statstcs & OR, Tel Avv Unversty, Takag, H.: Analyss of Pollng Systems. MIT Press, Takag, H.: Queueng analyss of pollng models: an update. In: Takag, H. (ed.) Stochastc Analyss of Computer and Communcatons Systems. North Holland, 1990, pp Yechal, U.: A new dervaton of the Khntchne-Pollaczek formula. In: Haley, K.B. (Ed.) Operatonal Research '75. North Holland, 1976, pp Yechal, U.: Optmal dynamc control of pollng systems. In: Cohen, J.W., Pack, C.D. (Eds.) Queueng, Performance and Control n ATM. North Holland, 1991, pp