Stability of the Greedy Algorithm on the Circle
SStability of the Greedy Algorithm on the Circle
Leonardo T. Rolla, Vladas Sidoravicius
Argentinian National Research Council, University of Buenos AiresNYU-ECNU Institute of Mathematical Sciences at NYU ShanghaiCourant Institute of Mathematical Sciences, New York University
Abstract
We consider a single-server system with service stations in each point of the circle. Cus-tomers arrive after exponential times at uniformly-distributed locations. The server movesat finite speed and adopts a greedy routing mechanism. It was conjectured by Coffman andGilbert in 1987 that the service rate exceeding the arrival rate is a sufficient condition for thesystem to be positive recurrent, for any value of the speed. In this paper we show that theconjecture holds true.
This preprint has the same numbering of sections, equations theorems and figures as the thepublished article “
Comm. Pure Appl. Math. 70 (2017): 1961–1986. ” In this paper we study a greedy single-server system on the unit-length circle R / Z . Customersarrive following a Poisson process with rate λ . Each arriving customer chooses a position on R / Z uniformly at random and waits for service. If there are no customers in the system, the serverstands still. Otherwise, the server chooses the nearest waiting customer and travels in that directionat speed v >
0, ignoring any new arrivals. Upon reaching the position of such customer, the serverstays there until service completion, which takes a random time T that is independent of the pastconfigurations and has expectation µ − .The above system was introduced by Coffman and Gilbert in 1987 [CG87], and since then becamea paradigm example of a routing mechanism that depends on the system state. This is the so-called greedy server , due to the simple strategy of targeting the nearest customer.Continuous-space models provide natural approximations for systems with a large number of ser-vice stations embedded in a spacial structure, and their description is usually more transparentthan the discrete-space formulation, mostly because the latter often is obscured by combinatorialaspects. However, systems with greedy routing strategies in the continuum are extremely sensitiveto microscopic perturbations, and their rigorous study represents a mathematical challenge.It was conjectured in [CG87] that the greedy server on the circle should be a stable system when λ < µ for any v >
0. Since then, a number of related models have been proposed and studied.Stability was verified under light-traffic assumptions, i.e., for λ and µ fixed and v large enough, a r X i v : . [ m a t h . P R ] S e p L. T. Rolla, V. Sidoraviciusand for the greedy server on a discrete ring Z /n Z . However, these approximations were unableto identify and tackle the main difficulty of this system, which is is due to the interplay betweenthe server’s motion and the environment of waiting customers that surround it. This interplay isgiven by the interaction resulting from the choice of the next customer and the removal of thosewho have been served. In this paper we prove stability for the greedy server. Definition.
We say that t is a regeneration time if the system becomes empty at time t , i.e.,if there is one customer at time t − and no customers at time t +. Let τ ø := inf { t > t is a regeneration time } . We say that the system is recurrent if, starting from the empty state ø,there will be a.s. a regeneration time, i.e., P ø [ τ ø < ∞ ] = 1. We say that the system is stable , or positively recurrent , if E ø [ τ ø ] < ∞ . Theorem 1.1.
Suppose that the distribution of the service time T is geometric, exponential, ordeterministic. For any λ < µ and any v > , the greedy server on the circle is stable.Remark. In our approach, it is crucial that the arrivals are Poisson in space-time. There is a dynamic version of the greedy server, where new arrivals are not ignored while the server istraveling. This variation might be studied by similar arguments, but the dynamic mechanismintroduces some extra complications that will not be considered here. A proof of stability forgeneral service times having an exponential moment follows from the same approach as presentedhere, requiring a little extra work due to the lack of Markov property. We present the proof forexponentially distributed service times with µ = 1. The cases of geometric or deterministic timesonly differ in notation.For the proof of Theorem 1.1, we consider a representation for the conditional distribution of theset of waiting customers in terms of a stochastic evolution of profiles. In this framework, the serverlearns only the information that is necessary and sufficient to determine the next movement, andthe positions of further waiting customers remain unknown. This approach was used in [FRS15]to show that the greedy server on the real line is transient, which is an important ingredient in ourproof of stability.In the remainder of this introduction, we review some known results on the greedy server andrelated models, discuss the problem of self-interaction, describe the approach based on a stochasticevolution of profiles, present a heuristic discussion in order to highlight the main ideas of the proof,and finally give a brief outline of the paper structure. Stability was verified for the greedy server on R / Z under light-traffic assumptions [KS96] and forthe greedy server on a discrete ring Z /n Z [FL96, FL98, MQ99], see below. It was also shown forseveral related models, including a class of non-greedy policies [KS94], a gated-greedy variant onconvex spaces [AL94], and random non-greedy servers on general spaces [AF97]. See [RNFK11]and references therein for a recent review.The light-traffic regime is given by λ (cid:18) µ + 12 v (cid:19) < . tability of the Greedy Algorithm on the Circle 3This regime was studied in [KS96], particularly the limit λ → v gives an upper bound for thetravel time between two consecutive services, since is the largest distance within the unit-lengthcircle. Adding this bound to the service time allows a comparison between the greedy server on R / Z and a stable M/G/ v than the above inequality allowsby obtaining a stochastic upper bound on the distance to the nearest customer better than . Butin any case, it may not be extended to general v >
0, because the presence of the server interferesseverely with the conditional distribution of the locations of remaining customers.On the other hand, stability under the general condition λ < µ is known to hold for the pollingserver on R / Z , i.e., the server whose strategy is to always travel in the same direction. In [KS92]this fact was proven using a decomposition of the set of waiting customers into a collection ofGalton-Watson trees that turn out to be subcritical for λ < µ . This decomposition provides adetailed description of the busy cycles (sequence of configurations observed between two consecutiveregeneration times) and the stationary state, but if one only wants to prove stability, there is asimple and robust argument. Take ε < − λµ and K such that Kµ + 1 v < (1 − ε ) Kλ .
The above inequality implies that, whenever the number of waiting customers N is larger than K ,the time it will typically take to serve the first N customers, including service and travel time, isless than the time it will typically take for the next N − εN arrivals, resulting in a net decreaseby εN on the number of waiting customers.Simulations indicate that, under heavy traffic conditions, the greedy server dynamics resemblesthat of the polling server [CG87]. This suggests that a possible strategy for proving stability ofthe greedy server might be to adapt the above argument. In this case, the first step would be tounderstand its local behavior, and a natural approach is to consider a system on an infinite line. Amodel on Z was studied in [KM97], where it is shown that the server is eventually going to movein a fixed random direction.Some direct attempts also include the study of a greedy server model on the finite ring Z /n Z ,which was shown to be stable in [FL96, FL98, MQ99]. Each of these references provides differentarguments under a variety of general assumptions.Yet discrete models have not been able to grasp the microscopic nature of the greedy mechanismin continuous space, neither on Z nor on Z /n Z , and there are major obstacles in extrapolating anyapproach based on a discrete approximation. This difficulty is due to the self-interaction of theserver’s path at the microscopic level , which takes place because the server’s trajectory influencesthe set of waiting customers and at the same time is determined by the latter. The main difficulty in studying greedy server systems in continuous spaces is due to the interplaybetween the server’s motion and the environment of waiting customers that surround it. This L. T. Rolla, V. Sidoraviciusinterplay is given by the interaction resulting from the greedy choice of the next customer and theremoval of those who have been served. The server’s path is self-repelling , since the removal ofalready served customers makes it less likely for the greedy server to take the next step back intothe recently visited regions.In some well-known examples of self-repelling motions, the self-interaction comes from an explicitprescription of the distribution of the next step in terms of the past occupation times. For theexcited random walks [BW03], perturbed Brownian motions [CPY98, CD99, Dav96, Dav99, PW97],and excited Brownian motions [RS11], whenever there is a drift, it is pushing the motion in a certainfixed direction. For the random walk avoiding its past convex hull [ABV03, Zer05] and the prudentwalk [BFV10, BM10], there is a growing forbidden region containing the previous trajectory, whichstrongly pushes the motion outwards.For our greedy server, and also for the true self-avoiding walk [T´ot95, T´ot99], the true self-repellingmotion [TW98], and the Brownian motion with repulsion [MT08], there is a mixture of informa-tion, and “self-repulsion” does not immediately imply “repulsion towards ∞ ”, since the particle isallowed to cross its past path, receiving contradictory signals from its left and right-hand sides. Infact, some of the latter models are recurrent and some are transient.It was clear since these models were introduced that they could not be treated via standard methodsand tools. A lot remains to be understood even in dimension d = 1, and, despite the existenceof a few disconnected techniques that have proved useful in particular situations, this rich field ofstudy lacks a systematic basis. The greedy server model has two particular features. Unlike most of the above models, here there isno direct prescription of how the past trajectory influences the future in terms of occupation times.Moreover, this evolution is time-inhomogeneous in the sense that customers keep accumulating,which yields an increasing bias towards least recently explored regions with decreasing traveledinter-distances after each service.
To address the issues mentioned in the above subsection, we consider a representation of thecustomers environment which reflects its randomness as perceived by the server.More precisely, we only want to learn the information that is necessary and sufficient to determinethe next movement, and the positions of further waiting customers should remain unknown. Eachtime the server has to scan the system state to determine the position of the next target, we acquireexactly two pieces of information: the presence of a customer at that position and the absence ofany other customer at smaller distances, as illustrated in Figure 1.1.The arrivals are represented by a space-time Poisson Point Process ν ⊆ ( R / Z ) × R , and in thisapproach one is ignoring the points of ν that have not yet influenced the server’s trajectory. One canthink of this scheme as re-sampling the set of waiting customers at each departure time, accordingto the appropriate conditional distribution. The latter is given in terms of the space-time regionwhere the configuration ν has not been revealed. In this setting, the state of the system is given by Except for the family of universality classes given by the Schramm-L¨owner Evolutions [Sch00], which include2-dimensional loop-erased random walk [LSW04a] and several other models [LSW04b, Smi01, Smi10]. tability of the Greedy Algorithm on the Circle 5the positions of the server and the current customer, plus the profile corresponding to the boundaryof this region where ν is unknown. The knowledge of this triplet determines the distribution of itsfuture without the need of any further information from the past, yielding a Markovian evolution.See Figure 1.1. If the server is busy most of the time, the system must be stable, since on average the service time issmaller than the inter-arrival time. The fundamental problem in showing stability is therefore thepossibility that the server spend a long time zigzagging on regions with low density of customersdue to a trapping configuration produced by the stochastic dynamics.For the analogous model on the real line, this cannot be the case: the server may zigzag for a finiteperiod of time, but it is bound to eventually choose a direction and head that way [FRS15].On the same grounds, since the greedy routing mechanism is local , this can neither be the case onthe circle – at least until the server realizes that it is not operating on the infinite line .Suppose we are given a configuration where the circle is crowded of waiting customers, and, fromthis point on, our goal is to alleviate this situation. We would like to say that, with high probability,after a short time the server will choose a direction and then cope with its workload as the pollingserver would. ? ?? ????? ?? ?? R R R Figure 1.1:
On the left, space-time evolution of the greedy server system: Points of ν correspond tocostumer arrivals and are represented by round points. The continuous path represents the server’s motion.For two different times we depicted the set of waiting customers with squared points. In the center, forthe same evolution, each of the contiguous areas in different tones of gray correspond to regions where ν is revealed at the moment when the server needs to know the new target customer. Each of these regionscontains the point corresponding to such customer at their lateral boundary, and no other points. On theright, the profile corresponds to the union of all the regions where ν has been revealed up to the time t corresponding to the dotted line. The configuration ν outside this region has been erased, since the server’spath up to time t gave no information about this configuration. The bold arrow indicates the positions ofboth the server and the current customer. L. T. Rolla, V. SidoraviciusThere are two situations where the server may feel that it is on the circle rather than on the line.First, if it arrives at a given point x for the second time after performing a whole turn on thecircle, it will encounter an environment that has been affected by its previous visit. This is notan actual problem, because if it happens it will imply that all the customers which were initiallypresent will have then been served, and typically the server will have served more customers thannew ones will have arrived.The second difference is what poses a real issue. The server has a tendency to go into regions thathave been least recently visited , since in these regions the average interdistance between customersis smaller, and they have bigger chance to attract the server via its greedy mechanism. This isindeed how transience is proved on R . Let us call the age of a point in space the measurement intime units of how recently it was visited by the server in the past. On the line, the age is minimalat the server’s position, and increases as we go further away from the server . The new regionsencountered thus become older and older, and the server surrenders to the fact that the clearedregions it is leaving behind cannot compete with the old regions ahead.However, this is not true on the circle: the age profile cannot increase indefinitely. This givesrise to the possibility of the following tricky scenario. Imagine that on a tiny region around somepoint x the system is much older than on any close neighborhood. When the server enters thisregion, it will take a very long time to finish with all the waiting customers. After finishing withall these customers tightly packed in space, there will no longer be a strong difference between theages ahead and behind the server, who may end up going back to the region that has just beencleared, invalidating the argument.We deal with this difficulty by making two key observations. First, the age of the points on thecircle is monotone in some sense: there is only one local minimum, located at the server’s position,and one local maximum x , and the age increases as we move from the server towards x . Second, ifthe above scenario effectively happens and the server changes direction, the new configuration maybecome worse in terms of the number of waiting customers, but will be better in the sense thatthis sharp peak in the age profile has been flattened. In order to say that the new configuration is“better” in this situation, we need to quantify “badness,” taking into account a trade-off betweendiminishing the overall workload and leveling this singular region with excessively high concentra-tion. This is achieved by considering a Lyapunov functional that combines the total number ofcustomers and the maximum local density. This paper is divided as follows. In Section 2, we give some definitions and notation used through-out the text, and describe the process evolution. In Section 3, we introduce the stochastic evolutionof profiles. In Section 4.1, we define an observable B that will serve as a Lyapunov functional,along with a stopping time T so that B T has a downwards drift, and finally prove Theorem 1.1.The proof of downwards drift is given in Section 4.2 by showing that the greedy server behavesmost of the time like a polling server via a coupling with a system on the infinite line, which isdone in Section 4.3. The latter is studied with a block construction in Section 4.4 and a renewalargument in Section 4.5.tability of the Greedy Algorithm on the Circle 7 The symbol (cid:52) means stochastic domination between random elements taking value on the samepartially ordered space. Define a ∧ b = min { a, b } , a ∨ b = max { a, b } , and [ a ] + = a ∨
0. The indicator that x ∈ J is denoted by J ( x ), and the indicator that the system is in a given state at time s isdenoted by state( s ). The complement of a set J is denoted by J c when the space where we takethe complement is clear.We consider the circle as the quotient space R / Z , and for x, y ∈ R we write x ∼ = y if x − y ∈ Z .Moreover, we identify classes of R / Z with their representants on R , and refer to the points ortheir representants without distinction unless mentioned otherwise. We denote arcs on the circleby [ x, y ) ⊆ R / Z , given by the projection of [ x, y ) ⊆ R for any pair of representants x, y ∈ R suchthat x (cid:54) y < x + 1. Analogously for open or closed arcs. We define the clockwise distance (cid:126)d by (cid:126)d ( x, y ) = y − x ∈ [0 , x, y ] = ( y, x ] c and (cid:126)d ( x, y ) = 1 − (cid:126)d ( y, x ). The distance on R / Z is given by d ( x, y ) = (cid:126)d ( x, y ) ∧ (cid:126)d ( y, x ). We say that f is increasing on [ x, y ] ⊆ R / Z if f is increasingon any lifting [ x, y ] ⊆ R with x (cid:54) y < x + 1; analogously for f nonincreasing, nondecreasing, etc. Evolution of the greedy server system
The state of the system at time t is described bythe triplet ( C t , S t , C t ). Here C t denotes the set of customers present at the system, S t denotes theposition of the server, C t ∈ C t denotes the position of the customer being served or targeted by theserver, and C t = C t = ø when the system is empty. The process ( C t , S t , C t ) t (cid:62) is a strong Markovprocess, whose stochastic evolution we describe now.At all times, t (cid:55)→ S t is continuous andd S t d t = V t := , S t = C t or C t = ø ,v, S t (cid:54) = C t , (cid:126)d ( S t , C t ) < (cid:126)d ( C t , S t ) , − v, S t (cid:54) = C t , (cid:126)d ( S t , C t ) (cid:62) (cid:126)d ( C t , S t ) , (2.1)in the sense of right derivative.There are three different regimes: moving when S t (cid:54) = C t ∈ R / Z , serving when S t = C t , and idle when C t = ø. While the system is idle , S and C remain unchanged until an arrival happens. Whilethe server is moving , the evolution of S obeys (2.1), and C remains constant. This regime lastsuntil service starts , i.e., until the time s when S s = C s . During service , the evolution of S is againgiven by (2.1), C also remains constant, and service finishes according to an exponential clock ofrate 1.The moments when service finishes will be called departure times . At departure times t , the newregime will be either moving or idle . First, the current customer is removed from the system: C t = C t − \ {C t − } . Then C t is chosen as the nearest waiting customer, if any: C t = arg min { d ( S t , x ) : x ∈ C t } , or C t = ø if C t = ø . (2.2)During the whole evolution, arrivals happen at rate λ . An arrival consists of adding to C a newpoint z chosen uniformly at random on R / Z , i.e., C t = C t − ∪ { z } . If C t − (cid:54) = ø, i.e., the server wasmoving or serving, this is the only change. If C t − = ø, i.e., the system was idle, then also C isupdated by C t = z and a.s. the new regime is moving. L. T. Rolla, V. Sidoravicius The process ( C t , S t , C t ) t (cid:62) may be constructed from two point processes: a Poisson Point Process ν ⊆ ( R / Z ) × R + with intensity λ · d x d t , each point corresponding to the arrival of a new customerat position x at time t , and the Poisson Point Process T ⊆ R + corresponding to possible departuretimes (each mark t ∈ T effectively corresponds to a departure time if a customer was being servedup to time t − and is ignored if the server was idle or moving). For u and w denoting functions on R / Z or constants, letΓ wu = { ( x, s ) : x ∈ R / Z , u ( x ) < s (cid:54) w ( x ) } ⊆ ( R / Z ) × R + . The σ -algebra F t = σ ( ν t , T t ), where ν t = ν ∩ Γ t and T t = T ∩ [0 , t ], contains all the informationabout arrivals and departures up to time t , and consequently about ( C s , S s , C s ) s ∈ [0 ,t ] .The process ( S t , C t ) t (cid:62) is not Markovian. Indeed, the conditional distribution of ( S s , C s ) s (cid:62) t given F t depends on both ( S t , C t ) and C t . Yet the only interaction between ( S , C ) and C is given by (2.2).Namely, at each departure time t , C t is queried about the nearest waiting customer C t , if any. Theposition of C t reveals that C t ∩ [ S t − z, S t + z ] = {C t } , where z = d ( S t , C t ) < , and on the otherhand it gives no information about the complementary set C t ∩ [ S t − z, S t + z ] c of waiting customers.In the sequel we discuss the conditional distribution of C t given ( S s , C s ) s ∈ [0 ,t ] , the role played bythis conditional law, and the evolution of this law itself. Markovianity without C t By the Markov property of ( C t , S t , C t ) t (cid:62) with respect to {F t } t (cid:62) we have that the conditional law of ( S s , C s ) s (cid:62) t satisfies L (cid:2) ( S s , C s ) s (cid:62) t (cid:12)(cid:12) F t (cid:3) = L (cid:2) ( S s , C s ) s (cid:62) t (cid:12)(cid:12) ( C t , S t , C t ) (cid:3) . Let G t = σ (cid:0) ( S s , C s ) s ∈ [0 ,t ] (cid:1) ⊆ F t . In the sequel we consider the triple (cid:0) L ( C t |G t ) , S t , C t (cid:1) and study its evolution.By the observations in the previous paragraph, the evolution ( S s , C s ) s ∈ [0 ,t ] gives information about ν ∩ ( R / Z ) × (0 , t ] in a very precise way. At each departure time s , the new C s is chosen as the pointof C s that is closest to S s . At these times, C s is given by S s ± z , where z is the smallest distance forwhich there is a point ( S s ± z, s (cid:48) ) with s (cid:48) ∈ (0 , s ] in ν , not considering the points that correspondto customers who have already left the system. This reveals a rectangle [ S s − z, S s + z ] × (0 , s ]where ν has no more points that will participate in the construction of ( C r ) r>s , and the law of C s outside [ S s − z, S s + z ] is not affected. For times r between s and the next departure time, C r is given by the union of C s and the Poisson arrivals corresponding to ν ∩ ( R / Z ) × ( s, r ]. For thetimes s when the system is in the idle state, the revealed rectangle is the whole R / Z × (0 , s ].Iterating this argument, by time t the configuration ν has been revealed on the region given by theunion of such rectangles. Since all these rectangles have their base on t = 0, their union is of theform Γ w t , where w t ( x ) denotes the maximal height among all the rectangles whose base containsthe point x . In other words, the value of w t ( x ) is the most recent among: the departure times s ∈ (0 , t ] such that x ∈ [ S s − z, S s + z ]; and the times s ∈ (0 , t ] when the system was idle. The setof waiting customers C t \ {C t } is thus determined by the configuration ν on the complementarytability of the Greedy Algorithm on the Circle 9region Γ tw t . Therefore, the conditional distribution of C t \{C t } given G t is that of an inhomogeneousPoisson process on R / Z , with local intensity at each point x given by λ (cid:2) t − w t ( x ) (cid:3) d x. In summary, L (cid:104) ( S s , C s ) s (cid:62) t (cid:12)(cid:12)(cid:12) G t (cid:105) = L (cid:104) ( S s , C s ) s (cid:62) t (cid:12)(cid:12)(cid:12) ( w t , S t , C t ) (cid:105) . Since the evolving region (Γ w t ) t (cid:62) is increased at departure times t by adding a rectangle to Γ w t − ,this rectangle being in turn determined by S t and C t , we have L (cid:104) ( w s , S s , C s ) s (cid:62) t (cid:12)(cid:12)(cid:12) ( w s , S s , C s ) s ∈ [0 ,t ] (cid:105) = L (cid:104) ( w s , S s , C s ) s (cid:62) t (cid:12)(cid:12)(cid:12) ( w t , S t , C t ) (cid:105) ;i.e., ( w t , S t , C t ) is a Markov process with respect to its natural filtration. In our framework, weshall consider u t = w t − t (cid:54) w , so that ( u t , S t , C t ) t (cid:62) is a time-homogeneous strong Markov process. Evolution of ( u t , S t , C t ) The law of the evolution ( u t , S t , C t ) t (cid:62) is given as follows. As before,the system may be in one of three regimes, determined by ( S t , C t ).While moving or serving , the evolution of S and C are given by the same rules as in the previoussection: C remains constant, S satisfies (2.1), and in the serving regime service finishes at rate1. We no longer have C to account for the whole set of waiting customers. Instead of randomlyadding new customers at rate λ , this information is now encoded in the function u ( x ), with therule d u t ( x )d t = − ∀ x ∈ R / Z , (3.1)which rather accounts for the time period when new customers have been arriving to the systemat location d x .At departure times , instead of choosing the nearest point in C t as in (2.2), we take what would benearest point in a realization of a Poisson Point Process on R / Z with intensity − u t ( x )d x . Moreprecisely, at the departure times the system goes through an instantaneous random transition,which may lead to either a moving or an idle state, as we describe below. Let 0 < E < ∞ and0 < U < t − . The meaning of E is that the measure of the interval that needsto be explored before finding a point is exponentially distributed, and U is important in decidingthe position of such point in the boundary of this explored interval. The total intensity of waitingcustomers potentially present in the system is given by A ( u ) = (cid:90) R / Z − λu ( x )d x. If E (cid:62) A ( u t − ), take C t = ø , idle . Otherwise, let 0 < z < be the unique number such that (cid:82) S t + z S t − z ( − λu t − )d x = E , let a = − u t − ( S t − z ), b = − u t − ( S t + z ), choose C t = (cid:40) S t − z, U ∈ (0 , aa + b ) , S t + z, U ∈ [ aa + b , , (3.2)and the new regime is moving . Finally, take u t ( x ) = (cid:40) u t − ( x ) · [ S t − z, S t + z ] c ( x ) , E < A ( u t − ) , , E (cid:62) A ( u t − ) . (3.3)The idle regime C = ø can only be achieved together with u ≡ R / Z . While the system is idle,the state ( u, S , C ) remains unchanged until the first customer arrival, which happens according toan exponential clock of rate λ . The arrival consists of letting C t = z , where z is chosen uniformlyon R / Z . Immediately after an arrival, a.s. the new regime is moving . Framework
A piecewise continuous, upper semi-continuous function u ( x ) (cid:54) R / Z is calleda potential . Note that the evolution described above can start from any given potential u andpoints S , C such that C (cid:54) = ø if u (cid:54)≡
0. For shortness, the triplet ( u, S , C ) will be denoted by U . Let P U denote the law of ( U t ) t (cid:62) starting from U at t = 0.We say that u is a proper potential if there exist x min , x max ∈ R / Z such that u is nondecreasing onthe arc [ x min , x max ] and nonincreasing on the arc [ x max , x min ], or if u is monotone on any arc notcontaining x max ; see Figure 3.1. Given a proper potential u , we say that U = ( u, S , C ) is a properstate if either u ( C ) = u ( x max ), or C = ø and u ≡ Proposition 3.1.
Starting from a proper state U , P U -a.s. the process ( U t ) t (cid:62) remains in properstates for every t > .Proof. When the system is idle, U is a proper state by definition. At departure times, accordingto (3.2)-(3.3) the position of C changes and u is updated by increasing its value to 0 on an arc S SS CC C = ø u ≡ x min x max x max R / Z R / ZR / Z uu Figure 3.1: Three examples of proper states .tability of the Greedy Algorithm on the Circle 11containing both the new and the old C . This transformation preserves the condition of U being aproper state. When the system state is either moving or serving, u evolves according to (3.1), thatis, the subtraction of a constant, which also preserves the proper state condition. (cid:3) Remark.
Although C t is not determined by ( S t , C t ), we have that C t = ø if and only if C t = ø, andit is thus sufficient to consider the process ( U t ) t (cid:62) in the study of positive recurrence, defined onpage 2. This is the approach used henceforth. The goal of this this section is to prove Theorem 1.1. In Section 4.1, we define a Lyapunovfunctional B and a stopping time T . We then state Proposition 4.1 about the downwards drift of B at time T and use it to prove Theorem 1.1. In Section 4.2, we prove Proposition 4.1 making useof Proposition 4.3, which states that the total time that the server spends traveling before time T is stochastically bounded. In Section 4.3, we prove Proposition 4.3 by coupling the system onthe circle with another one on the real line. In Section 4.4, we show that the latter has positiveprobability of being transient (and in fact ballistic) using a block construction. In Section 4.5,we conclude with a renewal argument relying on the uniform estimates obtained in the blockconstruction.We spell some formulae for later reference. η = 1 − λ, Ψ = 2 η − , ε = ηλ , δ = ε . (4.1)The reason for these definitions will become clear as they are used in the proof. Given a proper potential u , let N = N ( u ) = sup x ∈ R / Z − u ( x ) ,B = B ( u ) = A ( u ) + 4 εN ( u ) . Notice that the evolution of u is given by (3.1) when the state is moving or serving, at departuretimes it jumps upwards according to (3.3), and it remains constant when the state is idle. It thusfollows that u t + s (cid:62) u t − s ∀ s, t (cid:62) . (4.2)Since λ + 4 ε <
1, it follows from (4.2) that B ( u t + s ) (cid:54) B ( u t ) + s ∀ s, t (cid:62) . (4.3)Let B ∗ denote a finite number that will be fixed later. We claim that, for any proper state U with B ( u ) (cid:54) B ∗ , P U (cid:20) τ ø < v + 1 (cid:21) (cid:62) (cid:0) − e − (cid:1) exp (cid:16) − B ∗ − − v (cid:17) > . (4.4)2 L. T. Rolla, V. SidoraviciusTo see why the claim is true, consider the event that the server travels towards the nearest customer C , then finishes service within T < d = d ( S , C ) is at most , thisdeparture time happens at t (cid:48) = dv + T < v + 1, implying that τ ø < v + 1. The first term on theright-hand side corresponds to the probability that T <
1. The second term is a lower bound forthe the conditional probability that C t (cid:48) = ø given t (cid:48) , since the latter is given by e − A ( u t (cid:48) ) (cid:62) e − B ( u t (cid:48) ) which by (4.3) is bounded by e − B ( u ) − t (cid:48) , proving the claim.By (4.3) and (4.4), the proof of Theorem 1.1 reduces to showing thatsup (cid:110) E U (cid:2) τ { B (cid:54) B ∗ } (cid:3) : U proper state, B ( u ) < ¯ B (cid:111) < ∞ ∀ ¯ B < ∞ , (4.5)where τ { B (cid:54) B ∗ } = inf { t : B ( u t ) (cid:54) B ∗ } .Let U be a proper state such that B ( u ) > B ∗ . In the proof of (4.5), we study the behavior of B ( u t )at a particular stopping time T that is defined below.Define the sets U = (cid:8) x ∈ R / Z : u ( x ) < − N (cid:9) , (4.6) G t = (cid:8) x ∈ R / Z : u t ( x ) > − t (cid:9) . Since u is a proper potential, U must be either R / Z or an open arc. Notice that G = ∅ and by (4.2)we have that G t is nondecreasing in t . By (3.1) and (3.3), it may only increase at departure times t , by adding a closed arc containing S t and C t . Thus G t is always either ∅ , or all R / Z , or a closedarc containing S t .We define the following stopping times: T + = T + ( u ) = Ψ B ( u ) , T ◦ = T ◦ ( u ) = inf { t : G t ⊇ R / Z } , T (cid:103) = T (cid:103) ( u ) = inf { t : G t ⊇ U } , T = T ( u ) = T ◦ ( u ) ∧ T (cid:103) ( u ) ∧ T + ( u ) . (4.7)It follows from (4.3) that B ( u T ) (cid:54) B ( u ) + T (cid:54) B ( u ) + T + = (Ψ + 1) B ( u ) . (4.8)A few comments are in order. Normally, T is attained because the condition for T (cid:103) is attained.The deterministic time T + is a safety caution : it bounds the possible damage that is caused whenthis condition is not attained in due time. The presence of T ◦ in the definition of T is innocuousfrom a formal point of view, since T ◦ (cid:62) T (cid:103) . We write it to indicate that T (cid:103) may be attained intwo conceptually different situations: either because U is “crossed” by ( G s ) s (cid:62) , or because U ispartly taken by ( G s ) s (cid:62) from one direction and then from the other, in which case the whole circle R / Z is taken. See Figure 4.1. Proposition 4.1 (Downwards drift) . For any proper state U , P U (cid:0) B ( u T ) (cid:54) (1 − ε ) B ( u ) (cid:1) (cid:62) − ρ, (4.9) where ρ = ρ ( B ( u )) satisfies ρ ( B ) → as B → ∞ , and ε is defined in (4.1). tability of the Greedy Algorithm on the Circle 13Writing D ( · ) = log B ( · ) B ∗ , (4.8) and (4.9) imply that P U (cid:0) D ( u T ) (cid:54) D ( u ) − ε (cid:1) (cid:62) − ρ, D ( u T ) (cid:54) D ( u ) + Ψ , T (cid:54) T + = Ψ B ∗ e D . (4.10)We are going to use the following fact, whose proof is omitted. Lemma 4.2.
Let ( Y n ) n ∈ N be i.i.d. Bernoulli random variables with P ( Y = Ψ) = 1 − P ( Y = − ε ) = ρ. Write P s for the law of ( S n ) n ∈ N given by S n = s + Y + · · · + Y n , and define σ = inf { n : S n (cid:54) } .Then there exists ρ ∗ > such that E s [ σe Ψ σ ] < ∞ for any ρ (cid:54) ρ ∗ and s < ∞ .Proof of Theorem 1.1. We need to show (4.5). First we use Proposition 4.1 to fix the value of B ∗ with the property that ρ ( B ) (cid:54) ρ ∗ for any B > B ∗ .Let U be a proper state such that B ∗ < B ( u ) < ¯ B . We start with D = D ( u ) >
0. Considerthe stopping time T = T ( u ) defined by (4.7) and define D = D ( u T ). For the shifted process( U T + t ) t (cid:62) , consider the stopping time T = T ( u T ) and write D = D ( u T + T ). Analogously,once T , T , . . . , T n have been constructed, consider, for the shifted process ( U T + T + ··· + T n + t ) t (cid:62) ,the stopping time T n +1 = T ( u T + T + ··· + T n ), and write D n +1 = D ( u T + T + ··· + T n + T n +1 ). Let γ =inf { n : D n (cid:54) } .Taking s = D , it follows from (4.10) that( D n ∧ γ ) n =0 , , ,... (cid:52) ([ S n ∧ σ ] + ) n =0 , , ,... , (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (a) (b) (c) Gtt R / Z S t u u u U NN T g u T g T◦ u T◦ T + u T + Figure 4.1: Evolution of G t starting from a given potential u , with the new potential u T at thestopping time T . From left to right there are three pairs of graphs, each one embedded in thespace-time ( R / Z ) × R . In (a) we depict a typical example when T = T (cid:103) . In (b) we show an instancewhere T = T ◦ . Finally, in (c) there is an example where the server remains confined for a longtime, preventing the condition for T (cid:103) to be attained up to time T = T + . Each pair represents thesystem evolution and the resulting potential: the graph on the left shows the parametrized curves( S t , t ) t ∈ [0 , T ] and ( x, u ( x )) x ∈ R / Z , and on the right there is ( x, u T ( x ) + T ) x ∈ R / Z together with thepoint ( S T , T ).4 L. T. Rolla, V. Sidoraviciuswhence γ (cid:52) σ . Therefore we get τ { B (cid:54) B ∗ } Ψ B ∗ (cid:54) γ (cid:88) n =1 T n Ψ B ∗ (cid:54) γ (cid:88) n =1 e D n − (cid:54) γ exp (cid:20) max (cid:54) n<γ D n (cid:21) (cid:54) γe D +Ψ γ , whence by Lemma 4.21Ψ B ∗ E U (cid:2) τ { B (cid:54) B ∗ } (cid:3) (cid:54) E U (cid:2) γe D +Ψ γ (cid:3) (cid:54) E s (cid:2) σe s +Ψ σ (cid:3) (cid:54) E ¯ s (cid:2) σe ¯ s +Ψ σ (cid:3) < ∞ , where ¯ s = log ¯ BB ∗ . (cid:3) Write A = A ( u ), N = N ( u ), B = B ( u ), A (cid:48) = A ( u T ), N (cid:48) = N ( u T ), B (cid:48) = B ( u T ). We decomposetime in three parts: T = M + S + I , (4.11)where M = (cid:90) T moving( s )d s, S = (cid:90) T serving( s )d s, I = (cid:90) T idle( s )d s. By definition of T ◦ , the system cannot be idle for any t < T , thus I = 0. For each t >
0, let N t denote the number of departures times in (0 , t ]. Fix N = N T , the number of customers served upto time T . The total time spent with services during (0 , T ] is given by S = N (cid:88) n =1 T n + βT N +1 for some 0 (cid:54) β <
1, where ( T n ) n ∈ N are i.i.d. exponential random variables.Writing A t = A ( u t ), it follows from (3.1) that d A t d t = λ for Lebesgue-a.e. t < T . Moreover, ( A t ) t jumps downwards at departure times, and (3.3) reads as A t = (cid:2) A t − − E (cid:3) + . Since A t > t < T , A (cid:48) satisfies A (cid:48) = A + λ T − (cid:16)(cid:80)
N − n =1 E n + β (cid:48) E N (cid:17) , where 0 < β (cid:48) (cid:54) E n ) n ∈ N are i.i.d. exponential random variables.We now present the last ingredient, which is proved in the next subsection. Proposition 4.3 (Polling behavior) . The distribution of M under P U is tight: P U (cid:8) M > t (cid:9) −→ t →∞ uniformly over all proper states U . tability of the Greedy Algorithm on the Circle 15 Proof of Proposition 4.1.
It follows from Donsker’s invariance principle and from Proposition 4.3that (cid:80) N +1 n =1 T n < N + δB, (4.12) (cid:80) N − n =1 E n > N − δB, (4.13) M < δ Ψ B, (4.14)hold with high probability as B → ∞ , uniformly in U .Assume that (4.12), (4.13), and (4.14) happen. Putting these altogether yields0 (cid:54) A (cid:48) = A + λ T − (cid:80)
N − n =1 E n − β (cid:48) E N (cid:54) A + λ T − N + δB by (4 . (cid:54) A + λ T − S + 2 δB by (4 . A + λ T − T + M + 2 δB by (4 . (cid:54) A − η T + δ Ψ B + 2 δB by (4 . (cid:54) A − η T + 2 δ Ψ B since Ψ > < B − η T = η ( T + − T ) . since A < B, δ Ψ < T < T + . It then follows from the definition of T that U ⊆ G T ,whence u T ( x ) > −T for x ∈ U. But by (4.2) and the definition of U , we have u T ( x ) (cid:62) u ( x ) − T (cid:62) − N/ − T for x ∈ U c . Therefore, N (cid:48) (cid:54) N/ T . Combining this and (4.15): B (cid:48) − B (cid:54) ( − η T + 2 δ Ψ B ) + 4( T − N/ ε (cid:54) − (2 ε − δ Ψ) B − ( η − ε ) T since N > B (cid:54) − (2 ε − δ Ψ) B since 4 ε < η = − εB. (cid:3) In this section we prove Proposition 4.3 via a coupling with the greedy server on the real line. Thelatter eventually moves towards one of the two directions and spends little time going backwards,which was shown in [FRS15]. We consider a periodic extension of the initial potential u on thecircle, and approximate it by another potential with less oscillations, for which we can generalizethis result.The rules described in Section 3 may also be used to construct evolutions on the real line, whichwe will couple with the greedy server on the circle.Let us start with an informal description of how these systems are coupled.First, define an evolution on R by a lifting from R / Z : extend the potential periodically and replaceboth the server and the currently served customer by infinitely many replicas at unit interdistances.6 L. T. Rolla, V. SidoraviciusIf all the replicas evolve using the same randomness, this system remains periodic for all times.Moreover, one can recover the original system by projecting back from the line back onto the circle,so it is basically the same system.Now remove all the replicas and consider the system with a single server. For short times, thisserver evolves just like in the system with all the replicas. In fact, this remains true until the firsttime when the server needs to query for the presence of customers in a region that has alreadybeen queried by another replica. So the systems will uncouple at time T [1] defined below, or T ◦ forthe system on R / Z .Finally, this system with a single server on R can be coupled with another similar system alsoon R , which at t = 0 starts with the same positions for both the server and served customer, but aslightly different potential. If the same randomness is used for both systems, the servers will evolvetogether until the first time when they query for the presence of customers in a region where thepotential was initially different. This time of uncoupling is given by T U defined below. It typicallyoccurs before T [1] , in which case it will correspond to T γ on the circle.In summary, we can couple the system on the circle with one on the line, and the latter with anotherone having different initial potential, this double coupling lasting until T U ∧ T [1] . Figure 4.2 showsan example where T U is attained first, and Figure 4.1(b) shows an example on the circle where T [1] would be attained first.We now make the above description precise. Coupling with the greedy server on R A potential is a piecewise continuous, upper semi-continuous function ¯ u ( x ) (cid:54) R with (cid:82) R − ¯ u d x = ∞ . The evolution of ( ¯ U t ) t (cid:62) is defined in thesame way as on the circle, i.e., satisfying (2.1),(3.1),(3.2),(3.3).Let U be a proper state on the circle and ¯ u the periodic extension of u on R . Without loss ofgenerality, in the sequel we assume that S = 0. Take ¯ S = 0 and let ¯ C be the only representant of C in [ − , ).We define H t = (cid:8) x ∈ R : ¯ u t ( x ) > − t (cid:9) . By the same arguments as for the greedy server on R / Z , H t is nondecreasing in t ; it is empty untilthe first departure time, after which it consists of a closed interval containing both ¯ S t and ¯ C t .Define the stopping time T [1] = inf { t : | H t | (cid:62) } . For each t < T [1] , we define the map π t that takes each point x ∈ [ L ( t ) , L ( t ) + 1) ⊆ R to itsprojection x ∈ R / Z , where L ( t ) is chosen as follows. If H t = ∅ , we take L ( t ) = − ; otherwise if H t (cid:54) = ∅ , we take L ( t ) = inf H t . For a function w : R → R define π t w = w ◦ π − t .Recall from (4.6) that the set U is either the whole circle or an open arc not containing S = 0. Let¯ U = (cid:110) x ∈ R : ¯ u ( x ) < − N ( u )2 (cid:111) and take l = inf (cid:0) ¯ U ∩ [ − , (cid:1) , r = sup (cid:0) ¯ U ∩ [0 , (cid:1) . tability of the Greedy Algorithm on the Circle 17Finally, consider another initial state ˜ U given by ˜ S = ¯ S , ˜ C = ¯ C , and˜ u ( x ) = (cid:40) ¯ u ( x ) , x ∈ ( l, r ) , − N , otherwise . (4.16)Define the evolution ( ˜ U t ) t (cid:62) again by the same rules as for ¯ U , and consider the stopping time T U = inf (cid:8) t : H t (cid:54)⊆ ( l, r ) (cid:9) . Lemma 4.4 (Coupling) . The evolutions ( ¯ U t ) t (cid:62) and ( ˜ U t ) t (cid:62) on the line and ( U t ) t (cid:62) on the circlemay be constructed on the same probability space, satisfying T ◦ = T [1] , T (cid:103) = T U ∧ T [1] , U t = π t (cid:0) ¯ U t (cid:1) for all t < T [1] , ¯ S t = ˜ S t for all t < T U . Proof.
The coupling given by Lemma 4.4 is illustrated in Figure 4.2. The evolution of ( U t ) t can beconstructed using an i.i.d. sequence ( E n , U n , T n ) n , where E n and U n are the exponential uniformused as input for (3.2) and (3.3) at each departure time t n , and T n are the service times prior tothe n -th departure time. (When the system enters the idle state, another clock will be needed todetermine the next arrival time, but this state cannot be achieved before T ◦ .)The coupling is simple: we use the same sequence ( E n , U n , T n ) n to build ( ¯ U t ) t and ( ˜ U t ) t . It remainsto check that this coupling a.s. satisfies the identities stated in the lemma; the details are left tothe reader. (cid:3) Strong transience
For the evolution ( ˜ U t ) t , the total distance traveled by the server betweentimes t and t (cid:48) is denoted by V t (cid:48) t ( ˜ S · ) := (cid:90) t (cid:48) t | ˜ V s | d s = v (cid:90) t (cid:48) t moving( s )d s. We say that ( ˜ S t ) t (cid:62) is transient if, for each M >
0, sup (cid:8) t : ˜ S t ∈ [ − M, M ] (cid:9) < ∞ . If moreover (cid:12)(cid:12)(cid:12) ˜ S t − ˜ S (cid:12)(cid:12)(cid:12) (cid:62) V t ( ˜ S · ) for all t > , (4.17) ll rr S t ¯ S t ˜ S tu u u U Figure 4.2: Illustration of how U , ¯ U , and ˜ U evolve together until time T (cid:103) .8 L. T. Rolla, V. Sidoraviciuswe say that ( ˜ S t ) t (cid:62) is strongly transient . The latter means that the total displacement mustincrease linearly with the traveled distance.For 0 (cid:54) α (cid:54)
1, we say that ˜ U is α -unimodal if ˜ u attains its maximum on ˜ C , and˜ u ( x ) (cid:54) α · inf y ∈ [ ˜ C ,x ] ˜ u ( y ) , ∀ x > ˜ C , ˜ u ( x ) (cid:54) α · inf y ∈ [ x, ˜ C ] ˜ u ( y ) , ∀ x < ˜ C . (4.18)Notice that with α = 1 this is equivalent to ˜ u being nondecreasing on ( −∞ , ˜ C ] and nonincreasingon [ ˜ C , ∞ ), and for α < u (cid:54)
0, the condition is weaker.The result below is a consequence of Proposition 1 in [FRS15], written in our notations.
Proposition 4.5.
Given any ˜ U that is α -unimodal with α = 1 , ( ˜ S t ) t (cid:62) is a.s. transient. In order to prove Proposition 4.3, we shall obtain a slightly stronger result:
Proposition 4.6.
Given any ˜ U that is α -unimodal with α = , there exists a random time T Z satisfying P ˜ U ( T Z < ∞ ) = 1 , and such that ( ˜ S T Z + t ) t (cid:62) is strongly transient. Moreover, the numberof departure times ˜ N T Z before T Z is tight: P ˜ U (cid:110) ˜ N T Z > k (cid:111) −→ k →∞ uniformly over all α -unimodal ˜ U . Proposition 4.6 is proved in Sections 4.4 and 4.5 by adapting the multi-scale construction of [FRS15]to the case of α -unimodal initial states. Proof of Proposition 4.3.
We first observe that ˜ U , with ˜ u defined by (4.16), is α -unimodal for α = . By definition of M and V , M = 1 v V T ( S · ) (cid:54) v V T (cid:103) ( S · )and by Lemma 4.4, V T (cid:103) ( S · ) = V T [1] ∧T U ( ¯ S · ) = V T [1] ∧T U ( ˜ S · ) (cid:54) V T U ( ˜ S · ) . By definition of T U , we have that ˜ S t ∈ [ l, r ] ⊆ [ − ,
1] for all t < T U . The distance traveled by theserver between consecutive departure times is thus bounded by 2, and therefore V T Z ∧T U ( ˜ S · ) (cid:54)
2( ˜ N T Z ∧T U + 1) (cid:54)
2( ˜ N T Z + 1) . In case T U (cid:54) T Z , this upper bound for V T U ( ˜ S · ) is good enough. So consider the case T Z < T U andwrite V T U ( ˜ S · ) = V T Z ( ˜ S · ) + V T U T Z ( ˜ S · ) . By (4.17) and the definition of T Z , V T U T Z ( ˜ S · ) (cid:54) . Summarizing, M (cid:54) v (cid:16) N T Z (cid:17) and the result then follows from Proposition 4.6. (cid:3) tability of the Greedy Algorithm on the Circle 19 In the remainder of this section we give a short but self-contained proof of Proposition 4.6 using ablock argument. The reader will find a similar construction, with a more extended explanation ofthe main ideas, in [FRS15]. Since only ˜ U is concerned, we shorten notation and write U instead.Each time C or c appears, it denotes a different constant that is positive, finite, and depends onlyon v .Let A t = σ (( U s ) s ∈ [0 ,t ] ) denote the natural filtration for ( U t ) t (cid:62) . We construct a sequence ofstopping times 0 = L < L < · · · and define the corresponding events of success A j ∈ A L j +1 interms of U L j . The construction will have the following properties. For some sequence p j and any U that is α -unimodal, P U ( A j |A L j ) = P U ( A j |U L j ) (cid:62) p j on A ∩ · · · ∩ A j − , and (cid:89) j p j > . (4.19)The event ∩ ∞ j =0 A j implies strong transience of ( S t ) t (cid:62) . Almost surely, for each j = 0 , , , . . . , thestate of U L j is serving .We assume without loss of generality that the state of U is serving, that λ = 1, and that S = 0.Take σ = sgn S L to indicate the direction in which subsequent blocks are supposed to grow. Let Z j = σ S L j , N j = L j − u ( S L j ), Q j = N L j +1 − N L j , X j = Z j +1 − Z j , M j = L j +1 − L j .The triggering step j = 0 is defined as follows. We always take Q = 1, and the first step consistsof finishing with the customer that is being served at time L , then traveling towards the nearestcustomer at position σZ , and L is the stopping time attained as soon as the server reaches thisposition. The event A means success at the Step j = 0, and is defined by the following conditions: X − (cid:54) X (cid:54) X +0 and M − (cid:54) M (cid:54) M +0 , where X − = N , X +0 = 36 , M − = 1 , and M +0 = 2 + v . If there is no success, we declare Step 0 to have failed and stop. In the sequel we assume withoutloss of generality that σ = +1.For j (cid:62)
1, suppose that Steps 0 , , , . . . , j − u L j , and take (cid:96) j = (cid:100) j / (cid:101) , D j = (cid:96) j . Let s j, = S L j and s j, , s j, , . . . , s j,(cid:96) j , s j,(cid:96) j +1 denote the positions ofthe next (cid:96) j + 1 customers. Step j may be successful, which is denoted by the event A j , in twosituations: first, if s j,n > s j,n − for n = 1 , . . . , (cid:96) j , in which case we take Q j = (cid:96) j ; second, if thereis only one ˜ n ∈ { , . . . , (cid:96) j + 1 } such that ˜ n (cid:54) = (cid:96) j + 1 and s j, ˜ n < s j, ˜ n − , in which case we take Q j = (cid:96) j + 1. If none of these two happen, we declare Step j to have failed and stop. Otherwise,in either of the above two cases we say that Step j is successful if moreover Q − j (cid:54) Q j (cid:54) Q + j , X − j (cid:54) X j (cid:54) X + j , M − j (cid:54) M j (cid:54) M + j ,V L j +1 L j ( S · ) (cid:54) X j + D j N j , Z j − < S t < Z j +1 for L j (cid:54) t < L j +1 , where Q − j = (cid:96) j , Q + j = (cid:96) j + 1 , X − j = (cid:96) j − N j +1 , X + j = (cid:96) j N j , M − j = Q − j , M + j = 2 Q + j + X + j v . Here thetime L j +1 is given by the instant when the server reaches the last customer, located at Z j +1 , andthe next block starts with this customer being served.We now estimate the probability of success P U ( A j |U L j ) on A ∩· · ·∩ A j − by considering a numberof events that imply A j .0 L. T. Rolla, V. SidoraviciusFirst notice that A ∩ · · · ∩ A j − implies that M − j (cid:62) Cj / , N j = − u ( σZ j ) + L j (cid:62) L j (cid:62) M − + · · · + M − j − (cid:62) Cj / , X + j (cid:54) Cj − , X − j − (cid:54) Cj − , and thus M + j (cid:54) Cj / . Moreover, the conditionof U t being α -unimodal is preserved for all t . (Indeed, while the system state is serving or moving,by (3.1) the potential u (cid:54) C and increases the value of u to 0 on an interval that contains both the old andnew C . This increases the inf in (4.18), so this inequality still holds for x outside of such interval,whereas for x in such interval both u ( x ) and the inf become 0.) Thus, for j (cid:62)
1, the event A j − implies the following conditions on U L j : (cid:40) u L j ( x ) = u ( x ) − L j (cid:54) − N j for x > Z j ,u L j ( x ) (cid:62) − M j − for Z j − X − j − < x < Z j . (4.20)Let T , T , . . . , T (cid:96) j denote the service times of the customer being served at time L and the fol-lowing (cid:96) j customers. Let E , U , E , U , . . . , E (cid:96) j , U (cid:96) j , E (cid:96) j +1 , U (cid:96) j +1 be the exponential and uniformrandom variables used for determining the positions of s j, , s j,(cid:96) j +1 via (3.2).For j = 0, consider the event that 1 < T <
2, 36 < E <
72, and that U lies on the largestinterval among (0 , aa + b ) and [ aa + b , p = ( e − − e − )( e − − e − ) >
0. The requirement for U implies that u ( S L ) (cid:54) u ( − S L ).Hence, by α -unimodality of u , the occurrence of the above events imply36 < (cid:90) + z − z [ − u ( x ) + T ]d x (cid:54) (cid:90) + z − z max [ − z, + z ] ( T − u )d x (cid:54) − X [ u ( S L ) + T ] (cid:54) X N and 72 > (cid:90) + z − z [ − u ( x ) + T ]d x (cid:62) (cid:90) + z − z T d x (cid:62) z, which in turn imply the bounds on X and M , therefore P u ( A ) (cid:62) p > j (cid:62)
1, we observe that the positions s j, , s j, , . . . s j,(cid:96) j +1 can be sampled via a Poisson PointProcess on the region R = (cid:8) ( x, t ) ∈ R : Z j − X − j − (cid:54) x (cid:54) Z j + X + j , u L j ( x ) < t (cid:54) M + j (cid:9) , unless the elapsed time M j exceeds M + j or the exploration for customers leaves the in-terval [ Z j − X − j − , Z j + X + j ], which is ruled out a posteriori in case Step j is success-ful. This region R can be decomposed in a disjoint union R ∪ R , where R = (cid:8) ( x, t ) ∈ R : Z j (cid:54) x (cid:54) Z j + X + j , u L j ( x ) < t (cid:54) (cid:9) and R ⊆ [ Z j − X − j − , Z j ] × [ − M + j − , M + j ] ∪ [ Z j , Z j + X + j ] × [0 , M + j ].The inequalities in (4.20) and those preceding it imply that | R | (cid:54) ( X − j − + X + j )( M + j − + M + j ) (cid:54) Cj − / . The probability that there are two or more points in R is thus bounded by Cj − / .Define β ( x ) = (cid:82) xZ j [ − u L j ( z )]d z , x (cid:62) Z j , and write ( x , t ) , ( x , t ) , . . . the set of points found in R ,labeled by ordering x < x < x < · · · . Writing x = Z j , we have that ( β ( x n ) − β ( x n − )) n =1 ,...,(cid:96) j tability of the Greedy Algorithm on the Circle 21are i.i.d. exponential random variables. With positive probability, and tending to 1 exponentiallyfast in (cid:96) j as j → ∞ , both events23 ( (cid:96) j − (cid:54) β ( x (cid:96) j − ) (cid:54) β ( x (cid:96) j ) (cid:54) (cid:96) j and β ( x n ) − β ( x n − ) (cid:54) D j for n = 1 , , . . . , (cid:96) j occur. But since U L j is α -unimodal we have12 N j (cid:54) β ( x n ) − β ( x n − ) x n − x n − (cid:54) N j +1 . The above inequalities imply that X − j (cid:54) x (cid:96) j − − Z j (cid:54) x (cid:96) j − Z j (cid:54) X + j and x n − x n − (cid:54) D j N j = (cid:96) j N j (cid:54) X − j − . On the event that there are no points in R , we have V L j +1 L j = X j = x (cid:96) j − Z j . On theevent that there is one point in R , we have x (cid:96) j − − Z j (cid:54) X j (cid:54) x (cid:96) j − Z j and V L j +1 L j (cid:54) X j + 4 D j N j .This finishes the bounds on Q j , X j , V L j +1 L j , and ( S t ) t ∈ [ L j ,L j +1 ) .It remains to control M j , which is given by the sum T + · · · + T Q j − + v − V Lj +1 L j ( S · ) of servicetimes plus traveling time. The latter is non-negative and bounded by 2 X j /v , which is bounded by2 X + j /v . Therefore the inequality M − j (cid:54) M j (cid:54) M + j holds whenever Q j / < T + · · · + T Q j − < Q j ,which in turn occur with exponentially high probability in (cid:96) j .Finally, we use the bound on V L j +1 L j ( S · ) to prove strong transience. We add the requirement that V L j +1 L j ( S · ) = X j for j = 1. This changes the lower bound on probability of A , but it remainspositive. Notice that the same equality is true for j = 0 by construction. Now one can decompose V t ( S · ) in distances traveled in each of the two possible directions and again decompose thesedistances in the contribution from each block, and combine the bounds on X j − with D j N j (cid:54) X − j − to get V t ( S · ) (cid:54) S t . Having (4.19) in hands, we finally prove tightness of N T Z . We need that the probability of successin each block j not only be bounded from below by some p j but actually equal to p j . We introducean artificial coin toss to provide this last ingredient.Enlarge the underlying probability space to add an independent sequence of i.i.d. uniform variables˜ U j . For each j , define the event ˜ A j ⊆ A j by˜ A j = A j ∩ (cid:20) ˜ U j +1 < p j P ( A j |U L j ) (cid:21) . In words, we add an extra coin toss in order to have an exact equality instead of an upper bound: P U ( ˜ A j |U L j ) = p j on ˜ A ∩ · · · ∩ ˜ A j − . Notice that J = min { j : ˜ A j − does not occur } ∈ { , , , . . . } ∪ {∞} is a stopping time withrespect to { ˜ A j } j =0 , , ,... , where ˜ A j = σ ( A L j , ˜ U , ˜ U , . . . , ˜ U j − , ˜ U j ). The distribution of J is given2 L. T. Rolla, V. Sidoraviciusby P ( J > k + 1) = p p · · · p k . If J = ∞ , we have success for all j and ( S t ) t (cid:62) is stronglytransient, and we can take T Z = 0.If otherwise, J < ∞ , we have N L J (cid:54) Q + ( J ), where Q + ( j ) = Q +0 + Q +1 + · · · + Q + j . In this casewe can apply a time shift of L J and define ( U t ) t (cid:62) by U t = U t + L J . For this evolution ( U t ) t (cid:62) we can define the stopping times L , L , L , . . . , the events ˜ A , ˜ A , ˜ A , . . . , and the step of firstfailure J .By the strong Markov property, the conditional distribution of ( U t ) t (cid:62) given that J < ∞ is givenby P U LJ , and since U L J is α -unimodal, the conditional distribution of J given that J < ∞ isthe same: P ( J > k + 1 | J < ∞ ) = p p · · · p k .Again, if J = ∞ , ( S t + L J ) t (cid:62) is strongly transient and we take T Z = L J . Otherwise, N L J + L J (cid:54) Q + ( J ) + Q + ( J ). Analogously we can construct U , J , U , J , . . . until at some step K + 1 weget J K +1 = ∞ . We then take T Z = L J + L J + · · · + L KJ K , and we have that ( S t + T Z ) t (cid:62) is stronglytransient. As before, N T Z (cid:54) Q + ( J ) + Q + ( J ) + · · · + Q + ( J K ). But the distribution of the latterupper bound does not depend on U , and therefore N T Z is tight. Acknowledgments
We are grateful to S. Foss, who introduced us to this problem. We thank M. Jara for usefuldiscussions. This project had supported from grants PICT-2015-3154, PICT-2013-2137, PICT-2012-2744, PIP 11220130100521CO, Conicet-45955, MinCyT-BR-13/14 and MathAmSud-2014-LSBS.
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