Dynamic Vehicle Routing with Moving Demands Part I: Low speed demands and high arrival rates

Transcription

1 ACC 2009, Submitted St. Louis, MO Dynamic Vehicle Routing with Moing Demands Part I: Low speed demands and high arrial rates Shaunak D. Bopardikar Stephen L. Smith Francesco Bullo João P. Hespanha Abstract We introduce a dynamic ehicle routing problem in which demands arrie ia a temporal Poisson process with a certain arrial rate, and uniformly distributed along a line segment. Upon arrial, the demands moe in a fixed direction perpendicular to the line with a fixed speed. A serice ehicle, modeled as a first-order integrator with speed greater than that of the demands, seeks to sere these mobile demands. For the existence of any stabilizing serice policy, we determine a necessary condition on the arrial rate of the demands in terms of the problem parameters; (i) the speed ratio between the demand and serice ehicle, and (ii) the length of the line segment on which demands arrie. Next, we propose a noel serice policy for the ehicle that inoles sericing the outstanding demands as per the traeling salesperson path (t-tsp) through the moing demands. We derie a sufficient condition on the arrial rate of the demands for stability of the TSP-based policy, in terms of the problem parameters. We show that in the limiting case in which the demands moe much slower than the serice ehicle, the necessary and the sufficient conditions on the arrial rate are within a constant factor. We also proide an upper bound on the steady-state expected time spent by each demand before being sered. I. INTRODUCTION Dynamic ehicle routing problems such as the dynamic traeling repairperson problem (DTRP), consider one (or more) serice ehicles that seek to sere demands that arrie ia some spatio-temporal process in a region, and upon arrial the demands remain at their location until they are sered. In this work, we introduce a dynamic ehicle routing problem in which the demands moe with a specified elocity upon arrial, and we design policies for a single ehicle that seeks to sere them. This problem has applications in areas such as perimeter defense, wherein the demands could be isualized as moing targets trying to cross a region under sureillance by a UAV. Another application is in the automation industry where the demands are objects that arrie continuously on a coneyor belt and a robotic arm seeks to perform a pick-and-place operation on them. The DTRP was first introduced in [1] in which the goal is to minimize the expected time spent by each demand before being sered. In [1] the authors propose a policy that is optimal in the case of low arrial rate, and seeral policies within a constant factor of the optimal in the case of high arrial rate. In [2], they also study multiple serice ehicles, and ehicles with finite serice capacity. In [3], a single policy is proposed which is optimal for the case of low arrial rate and performs within a constant factor of the The authors are with the Center for Control, Dynamical Systems and Computation, Uniersity of California at Santa Barbara, Santa Barbara, CA 93106, USA, {shaunak,stephen,bullo}@engineering.ucsb. edu, hespanha@ece.ucsb.edu best known policy for the case of high arrial rate. In [4], decentralized policies are deeloped for the multiple serice ehicle ersions of the DTRP. The Euclidean traeling salesperson problem (ETSP) consists of determining the minimum length tour through a gien set of points in a region [5]. Vehicle routing with targets moing on straight lines was introduced in [6], where a fixed number of targets moe in the same direction with fixed speed, and the problem is to catch the maximum number of targets before they cross a finish line. A ariation of this problem with target motion on piece-wise straight line paths and with arying target speeds has been addressed in [7]. For the case in which there is no finish line, termed as the translational traeling salesperson problem (t-tsp), a polynomial-time approximation scheme has been proposed in [8] to catch all targets in minimum time. Other ariants of the ETSP in which the points are allowed to moe in different directions hae been addressed in [8] and in [9]. We introduce a dynamic ehicle routing problem in which demands arrie ia a temporal Poisson process with rate λ, and uniformly randomly on a line segment of finite length W. Upon arrial, the demands moe in a fixed direction perpendicular to the line and with a fixed speed < 1. A serice ehicle, modeled as a first-order integrator with unit speed, seeks to sere these mobile demands. Our main contributions are as follows: First, we show that to ensure the existence of a stabilizing policy, i.e., a finite expected time spent by a demand in the enironment, we must hae λ 4/W. Second, we propose a noel serice policy which inoles sericing all of the outstanding demands as per the translational traeling salesperson path (TSP) through them. We show that a sufficient condition for stability of this TSP-based policy is λ < (1 2 ) 3/2 /2W(1+) 2. For this policy, we also obtain an upper bound on the steadystate expected time a demand spends in the enironment before being sered. As the arrial rate λ +, the necessary stability condition implies that the demands must hae 0 +. This regime of low demand speed with high arrial rate is the focus of this paper. In this regime, the necessary and the sufficient stability conditions on the arrial rate mentioned aboe are within a constant factor. In companion paper [10], we analyze a first-come-firstsered (FCFS) policy in which the demands are sered in the order of their arrial. We show that in the regime of λ 0 +, the FCFS policy minimizes the expected time spent by a demand before being sered; while in the regime of 1, the FCFS is the optimal policy. Thus, for low demand speeds the TSP-based policy can stabilize higher arrial rates, while

2 for high demand speeds the FCFS can stabilize higher arrial rates. This is summarized in Fig. 1. Arrial rate λ e f a TSP stable TSP and FCFS stable Stabilization impossible for any policy FCFS stable (0, 0) Demand speed (1, 0) Fig. 1. A summary of stability regions for the TSP-based policy and the FCFS policy. Stable serice policies can exist only for the region under the dotted line. Cure a-b is due to Theorem IV.1, and cure c-f is due to Theorem V.1. Cures b-c, c-d and d-e, are established in [10]. This paper is organized as follows: we begin with background results on the traeling salesperson problems in Section II. The problem formulation is presented in Section III. The necessary condition for stability is deried in Section IV. The TSP-based serice policy and the main results are presented in Section V. Simulation results are presented in Section VI. II. BACKGROUND RESULTS ON TRAVELING SALESPERSON PROBLEMS In this section we reiew seeral results on determining shortest paths through sets of points. In what follows, gien a set of points in the plane, we are interested in the length of a path through the points that is not closed. These results will be applied in the analysis in Section V. A. The Euclidean Traeling Salesperson Path (ETSP) We are interested in the following Euclidean TSP problem. Gien n static points placed in R 2, determine the length of the shortest path through all the points. An upper bound on the length of such a path for points in a unit square was gien by Few [11]. Here we extend Few s bound to the case of points in a rectangular region. For completeness, we hae included the proof in the Appendix. Lemma II.1 (Euclidean TSP length) Gien n points in a 1 h rectangle in the plane, where h R >0, there exists a path that starts from one of the edges of the rectangle, passes through each of the n points exactly once, and terminates on the opposite edge with length upper bounded by 2hn + h + 5/2. We will also require a result on the length of a path through a large number of points. Gien a set Q of n points in R 2, b d c let ETSP(Q) denote the minimum length of a path through all the points in Q. The following is an established result. Theorem II.2 (Asymptotic TSP length, [12]) If n points are distributed independently and identically uniform in a compact region of area A, then there exists a constant β TSP such that ETSP(Q) lim = β TSP A. (1) n + n The current best estimate of the constant is β TSP , [13]. B. The Translational Traeling Salesperson Path Next, we describe the translational TSP which was proposed and soled in [8]. This problem is posed as follows. Gien a start point s(t), a set of points Q(t) := {q 1 (t),...,q n (t)} and a finish point f(t) all moing with the same constant speed and in the same direction, determine a salesperson path that starts from s, isits all points in Q and ends at f, and the length L T (s, Q,f) of which is minimum. A solution for this problem is the Conert-to-Static TSP method: (i) Define the map f : R 2 R 2 by ( x f (x,y) =, y ) (ii) Compute the static TSP that starts at f (s), passes through the set of points gien by {f (q 1 ),...,f (q n )} and ends at f (f). For this method, the following result is established. Lemma II.3 (Translational TSP length, [8]) The of the translational TSP is L T (s, Q,f) = L E (s, Q,f) + (y f y s ) 1 2, length where L E (s, Q,f) denotes the length of the static TSP with starting point f (s) := f (x s,y s ), passing through the set of points {f (q 1 ),...,f (q n )}, and ending at f (f) := f (x f,y f ). In other words, the length of the translational TSP is optimal if and only if the length of the TSP in the corresponding static instance is optimal. III. PROBLEM FORMULATION We consider a single serice ehicle that seeks to serice mobile demands that arrie ia a spatio-temporal process on a line segment with length W along the x-axis, termed the generator. The ehicle is modeled as a first-order integrator with speed upper bounded by one. The demands arrie uniformly distributed on the generator ia a temporal Poisson process with intensity λ > 0, and moe with constant speed < 1 along the positie y-axis, as shown in Fig. 2. We assume that once the ehicle reaches a demand, the demand is sered instantaneously. The ehicle is assumed to hae unlimited fuel and demand sericing capacity.

3 (0, 0) (X(t), Y (t)) Fig. 2. The problem set-up. The thick line segment is the generator of mobile demands. The dark circle denotes a demand and the square denotes the serice ehicle. W We define the enironment as E := [0,W] R 0 R 2, and let p(t) = [X(t),Y (t)] T E denote the position of the serice ehicle at time t. Let Q(t) E denote the set of all demand locations at time t, and n(t) the cardinality of Q(t). Sericing of a demand q i Q and remoing it from the set Q occurs when the serice ehicle reaches the location of the demand. A static feedback control policy for the system is a map P : E 2 E R 2, assigning a commanded elocity to the serice ehicle as a function of the current state of the system: ṗ(t) = P(p(t), Q(t)). Let D i denote the time that the ith demand spends within the set Q, i.e., the delay between the generation of the ith demand and the time it is sericed. The policy P is stable if under its action, lim E[D i] < +, i + i.e., the steady state expected delay is finite. Equialently, the policy P is stable if under its action, lim t + E[n(t)] < +, that is, if the ehicle is able to serice demands at a rate that is on aerage at least as fast as the rate at which new demands arrie. In what follows, our goal is to design stable control policies for the system. IV. A NECESSARY CONDITION FOR STABILITY In this section we proide a necessary condition on the arrial rate for the existence of a stabilizing policy. We begin by stating the main result of the section, with the remainder of the section dedicated to its proof. Theorem IV.1 (Necessary condition for stability) A necessary condition for the existence of a stabilizing policy is that λ 4 W. Before proing this result we state one of its key consequences. Corollary IV.2 (Constant fraction serice) A necessary condition for the existence of a policy which serices a fraction c (0,1] of the demands is that λ 4 c 2 W. Thus, for a fixed < 1 no policy can serice a constant fraction of the demands as λ +. To proe Theorem IV.1 we begin by looking at the distribution of demands in the serice region. Lemma IV.3 (Poisson point process) Suppose the generation of demands commences at time 0 and no demands are sericed in the interal [0,t]. Let Q denote the set of all demands in [0,W] [0,t] at time t. Then, gien a compact region R of area A contained in [0,W] [0,t], e λa ( λa) n P[ R Q = n] =, where λ := λ/(w). Proof: Let R = [l,l+ l] [h,h+ h] be a rectangle contained in [0,W] [0,t] with area A = l h. Let us calculate the probability that at time t, R Q = n (that is, the probability that R contains n points in Q). We hae P[ R Q = n] = P i=n [ i demands arried in [ h, h + h ]] P[n of i are generated in [l,l + l]]. Since the generation process is temporally Poisson and spatially uniform the aboe equation can be rewritten as P[ R Q = n] = Now, P[i demands arried in [0, h/]] i=n P[n of i are generated in [0, l]]. (2) P[i demands arried in [0, h/]] = e λ h/ (λ h/) i, i! and, ( )( ) n ( i l P[n of i are in [0, l]] = 1 l ) i n. n W W So, letting L := l/w and H := h/, and substituting in the aboe expressions, Eq. (2) becomes P[ R Q = n] = e λh L n (λh) i ( ) i (1 L) i n. i! n i=n (3) Rewriting (λh) i as (λh) n (λh) n i, and using the definition ( ) i i! = n (i n)!, we can write Eq. (3) as λh (λlh)n P[ R Q = n] = e (λh(1 L)) j j=0 λh+λh(1 L) (λlh)n = e λlh (λlh)n = e. Finally, since LH = A/(W), we obtain P[ R Q = n] = e λa ( λa) n, j!

4 where λ := λ/(w). Thus, the result is established for rectangles. Howeer, eery compact region can be written as a countable union of rectangles, and thus the result holds for eery compact, measurable region contained in [0,W] [0,t]. Remark IV.4 (Uniformly distributed demands) Lemma IV.3 shows us that the number of demands in an unsericed region is Poisson distributed with rate λ/(w), and conditioned on this number, the demands are distributed uniformly. We now establish a result on the expected time to trael from a demand to its nearest neighbor. For this we require a result on catching a demand in minimum time (cf. Fig. 3). Proposition IV.5 (Minimum time control, [14]) Gien the locations p := (X,Y ) E and q := (x,y) E at time t of the ehicle and a demand, respectiely, then the motion of the ehicle towards the point (x,y + T), where (1 2 )(X x) T(p,q) := 2 + (Y y) 2 (Y y) , minimizes the time taken by the ehicle to reach the demand. (0, 0) Fig. 3. p = (X,Y ) W C = (x, y + T) q = (x, y) Illustration of Proposition IV.5. Lemma IV.6 (Trael time bound) Consider the set Q of demands in E at time t. Let T d be a random ariable giing the minimum amount of time required to trael to a demand in Q from a ehicle position (X,Y ), selected a priori. Then E[T d ] 1 W 2 λ. Proof: To obtain a lower bound on the minimum trael time we can assume that Q contains many demands (i.e., t is ery large), and no demands hae been sericed. Consider a demand in Q with position (x, y) at time t. Using Proposition IV.5, we can write the trael time T from p := (X,Y ) to q := (x,y) implicitly as T(p,q) 2 = (X x) 2 + ((Y y) T(p,q)) 2. (4) Thus, we can define the set S T, such that any demand in S T can be reached from (X,Y ) in T time units. From Eq. (4) we see that the set S T is a circle of radius T centered at X,Y T. That is, S T := {(x,y) E : (X x) 2 + ((Y T) y) 2 T 2 }, where we hae omitted T s dependence on p and q. If the set S T does not intersect a boundary of E it has area πt 2, but in general its area is S T πt 2. Now, by Lemma IV.3 the demands in an unsericed region are uniformly randomly distributed with density λ = λ/(w). Let us compute the distribution of T d := min q Q T(p,q). For eery ehicle position p chosen before the generation of demands, the probability that T d > T is gien by P[T d > T] = P[ S T Q = 0] e λ S T e λπt2 /(W). Hence we hae E[T d ] = + 0 P[T d > T]dT + π 2 λπ/(w) = 1 W 2 λ. 0 e λπt2 /(W) dt We can now proe Theorem IV.1. Proof: [Proof of Theorem IV.1] A necessary condition for the stability of any policy (see, for example [1]) is that λe[t] 1, where E[T] is the steady-state expected trael time between demands i and i + 1. For eery policy E[T] E[T d ] 1 W 2 λ. Thus a necessary condition for stability is that λ 1 W 2 λ 1 λ 4 W. Finally, Corollary IV.2 follows since in order to serice a fraction c we require that cλe[t d ] < 1. V. THE TSP-BASED POLICY AND THE MAIN RESULT In this section, we present a noel serice policy for the ehicle which is based on computation of the translational traeling salesperson path (TSP) path through successie groups of outstanding demands. A. The TSP-Based Serice Policy and the Main Result The TSP-based serice policy is as follows: TSP-based serice policy Assumes: Serice ehicle has initial position (X, Y ), and all demands hae lower y-coordinates. 1 if no outstanding demands in the enironment then 2 Moe towards the generating line for a time interal of Y/(1 + ). 3 else 4 Let V be a irtual demand located at (X,0) moing with speed in the positie y-direction. 5 Serice all the outstanding demands by following a translational TSP starting from (X, Y ), and terminating at irtual demand V. 6 Repeat.

5 (X,Y ) V W Fig. 4. The TSP-based policy. The ehicle seres all outstanding demands inside the shaded rectangular region R(X, Y ) as per the translational TSP that begins at (X, Y ) and terminates at the irtual demand V. An instance of this policy is illustrated in Fig. 4. The TSPbased serice policy gies the following result. Theorem V.1 (TSP-based policy) (i) The TSP-based policy is stable if λ < (1 2 ) 3/2 2W(1 + ) 2, and, (ii) assuming that the TSP-based policy is stable, the steady state expected time spent by a demand in the enironment is no larger than ( ) 5W /(1 + ). 2Wλ/(1 2 ) 3/2 Proof: Let R(X,Y ) denote the region [0,W] [0,Y ] defined by the position (X,Y ) of the serice ehicle, as shown in Fig. 4. Obsere that at the end of eery iteration of this policy, all outstanding demands hae their y-coordinates less than or equal to that of the ehicle, and hence would be contained in R(X,Y ). Let the ehicle be located at p(t i ) = (X(t i ),Y (t i )) at time instant t i. If there are no outstanding demands in R(X(t i ),Y (t i )), then Y (ti) 1+ is the distance that the ehicle moes towards the generator. Thus, we hae Y (t i+1 ) = Y (t i ) Y (t i) 1 + = Y (t i) 1 +, if there are no unsericed demands in R(X(t i ),Y (t i )) at time t i. Otherwise, if there exist unsericed demands {q 1,...,q ni } where n i 1, in R(X(t i ),Y (t i )), then we hae Y (t i+1 ) = L T (p(t i ), {q 1,...,q ni },V (t i )), where L T (p(t i ), {q 1,...,q ni },V (t i )) is the time taken for the ehicle as per the translational TSP that begins at p(t i ), seres all n i demands and ends at the irtual demand V (t i ). Since the distribution of the demands inside R(X(t i ),Y (t i )) is spatially Poisson (cf. Lemma IV.3 from Section IV), we hae Y (t i+1 ) = Y (t i) 1 +, w.p. e λa, = L T (p(t i ), {q 1 },V (t i )), w.p. ( λa)e λa, = L T (p(t i ), {q 1,q 2 },V (t i )), w.p. ( λa) 2 e λa, 2! and so on, where A = WY (t i ) is the area of R(X(t i ),Y (t i )). We now seek an upper bound for the length L T (p(t i ), {q 1,...,q ni },V (t i )) of the translational TSP for which we use the Conert-to-Static TSP method (cf. Section II-A). For n i = k 1, inoking Lemma II.3 and writing Y i := Y (t i ), for conenience, L T (p(t i ), {q 1,...,q k },V (t i )) = L E (p(t i ), {q 1,...,q k },V (t i )) + (y V (t i) Y i ) 1 2 = L E (p(t i ), {q 1,...,q k },V (t i )) Y i 1 2 2WY i k (1 2 ) + Y i 3/ W 2 1, 2 where the second equality is due to y V (ti) = 0, and the inequality is obtained using Lemma II.1. Thus, we hae E[Y i+1 Y i ] Y i e λa ( 2WY i k (1 2 ) + Y i 3/ W ) ( λa) k 2 e λa, 1 2 k! k=1 where λ = λ/w from Lemma IV.3. Collecting the terms with Y i /(1 + ) together, we obtain E[Y i+1 Y i ] Y i 1 + ( λa) k e λa + k! k=0 ( 2 2 WY i k 5W ) ( λa) k + (1 2 ) 3/2 2 e λa 1 k=1 2 k! = Y i W (1 2 ) E[ ] 5W(1 e λa ) n 3/2 i Y i Y i Y i W Yi E [ ] 5W ni Y (1 2 ) 3/2 i Y i W Yi E[ni Y (1 2 ) 3/2 i ] + 5W = Y i W λwyi Yi (1 2 ) 3/2 W + 5W = Y i W λ (1 2 ) 3/2 Y i + 5W 2 1, 2 where the inequality in the fourth step follows by applying Jensen s inequality to the conditional expectation and the equality in the fifth step is due to the arrial process being spatially Poisson (cf. Lemma IV.3). Using the law of iterated expectation, we hae E[Y i+1 ] = E[E[Y i+1 Y i ]] 1 + E[Y 2λW i] + (1 2 ) E[Y 5W i] + 3/2 2 1, (5) 2

6 which is a linear recurrence in E[Y i ]. Thus, lim i + E[Y i ] is finite if Wλ (1 2 ) < 1 λ < (1 2 ) 3/2 3/2 2W(1 + ) 2. Thus, if λ satisfies the condition aboe, then expected number of demands in the enironment is finite and the TSPbased policy is stable. We now compute an upper bound on the steady state expected time a demand spends in the enironment. If we denote a := Wλ, b := 5W (1 2 ) 3/2 2 and Ȳ := 1 2 lim i + E[Y i ], then the recurrence Eq. (5) implies Ȳ b 1 a. Thus, in the steady state, the ehicle would be at a distance of at most b/(1 a) from the generator in expected alue. Suppose the jth demand arried between time iterations i 1 and i, i.e., in the time interal (t i 1,t i ]. Then, the distance traeled by demand i before being sericed is at most Y i+1, assuming that it is the first demand to arrie in the interal (t i 1,t i ] and the last among them to be sericed. Thus, the time spent by the jth demand in the enironment (i.e., the delay) satisfies D j Y i+1 /. Taking expectation, as j +, we must also hae i + ; otherwise, if i i < +, then it would mean that there are infinite number of demands which arrie in (t i 1,t i ], contradicting the fact that the system is stable. Thus, the steady state expected time spent by a demand in the enironment satisfies lim E[D j] j + lim E[Y i+1]/ = Ȳ / i + b 1 a, which is bounded if λ < (1 2 ) 3/2 /2W(1 + ) 2. B. Limiting Case of Low Speed Demands In this section we focus on the case when λ + and, by the necessary stability condition in Theorem IV.1, 0 +. Recall that for this case, the sufficient stability condition for the TSP-based policy is that λ < 1/(2W). This differs by a factor of 8 from the policy independent necessary stability condition of λ < 4/(W). By utilizing the tight asymptotic expression for the length of the TSP path, gien in Theorem II.2, in place of the bound in Lemma II.1, we can reduce this factor to approximately 2. To begin, consider an iteration i of the TSP-based policy, and let Y i > 0 be the position of the serice ehicle. Then, in the limit as λ +, the number of outstanding demands in that iteration n i +. Thus, applying Theorem II.2 and Lemma II.1, the position of the ehicle at the end of the iteration is gien by Y i+1 = β TSP ni A = β TSP ni Y i W, where A := Y i W is the area of the region below the ehicle at the ith iteration. Thus, conditioned on Y i being bounded away from 0, we hae E[Y i+1 Y i ] = β TSP E [ Wn i Y i ] β TSP WYi E[n i ], where we hae applied Jensen s inequality. Using Lemma IV.3, E[n i ] = WY i λ/(w) and thus E[Y i+1 Y i ] β TSP W 2 Y 2 i λ W = β TSP λwyi. Thus, the sufficient condition for stability of the TSP-based policy as λ + (and thus 0 + ) is λ < 1 β 2 TSP W 2 W, where β TSP Hence, in the limiting regime as λ +, the sufficient stability condition for the TSP-based policy is within a constant factor of the policy independent necessary condition. VI. SIMULATIONS In this section we present results of numerical experiments of the TSP-based policy. The linkern 1 soler was used to generate approximations to the optimal TSP tour. We use these experiments to study how the steady-state expected delay aries with the generation rate of the demands for different speeds. The policy was simulated for a fixed number of iterations, large enough to ensure that steady-state was reached. We obtained the steady-state delay by computing the mean of the delay in the last 20 iterations. This was repeated 10 times to obtain an estimate of the steady-state expected delay for one alue of the generation rate and for a gien speed. We then repeat the experiment for three different alues of the speed. The ariation of the expected delay with the generation rate is presented in Fig. 5 for different alues of speed. We obsere that for sufficiently large alues of such as = 0.2, the necessary condition on the arrial rate is almost 4 times that of the arrial rate that causes instability. Howeer, this ratio decreases as reduces. Specifically, for = 0.05, the ratio is approximately 2.5. These obserations are consistent with our theoretical analysis that predicts that in the limit as 0 +, the ratio of the necessary to sufficient condition on λ tends to 2. VII. CONCLUSIONS AND FUTURE DIRECTIONS We introduced a ehicle routing problem in which a serice ehicle seeks to sere demands that arrie ia a Poisson process on a line segment and that moe with a fixed speed in a direction perpendicular to the line. For the existence of a stabilizing serice policy, we first deried a necessary condition on the arrial rate of the demands as a function of the speed ratio between the demands and the ehicle, and 1 linkern is written in ANSI C and is freely aailable for academic research use at

7 Expected serice time = 0.05 = 0.1 = Generation rate λ Fig. 5. Simulation results for TSP-based policy: ariation of the steady-state expected delay with the arrial rate for three different alues of the demand speeds. A dotted line is an asymptote to a cure, i.e., the extrapolated alue of λ that leads to instability for a gien. The length of the generator is W = 50. the length of the line segment. Then, we proposed a noel serice policy for the ehicle which inoles sequentially sericing all the outstanding demands as per the translational TSP through the moing demands. We deried a sufficient condition on the arrial rate of the demands for stability of the TSP-based policy. In the limiting case of the relatie speed tending to zero, we showed that the necessary and the sufficient conditions on the arrial rate are within a constant factor. We also proided an upper bound on the expected time spent by the demand in the enironment before being sered. In companion paper [10], we analyze the first-comefirst-sered (FCFS) policy and show that in the regimes of high demand speeds, the policy is stable for higher arrial rates than the TSP-based policy. Further, we show that in the high demand speed regime, the FCFS is the optimal policy. In the future, we enision to address ersions of the present problem inoling multiple serice ehicles. Another interesting direction is to consider non-uniform spatial arrial of the demands on the generating line. ACKNOWLEDGMENTS This material is based upon work supported in part by ARO-MURI Award W911NF , ONR Award N and by the Institute for Collaboratie Biotechnologies through the grant DAAD19-03-D-0004 from the U.S. Army Research Office. REFERENCES [1] D. J. Bertsimas and G. 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Martin, Finite size and dimensional dependence of the Euclidean traeling salesman problem, Physical Reiew Letters, ol. 76, no. 8, pp , [14] R. Isaacs, Differential games. New York: John Wiley, APPENDIX In this Appendix we present the proof of Lemma II.1. Proof: Suppose the rectangular region is gien by 0 x 1, 0 y h. Let m be a positie integer (to be chosen later) and let the n points be denoted by {q 1,...,q n }. We now construct two paths through the points. The first consists of (a) the m + 1 lines y = 0,h/m,2h/m,...,h; (b) the n shortest distances from each of the n points to the nearest such line, each traeled twice, and (c) suitable portions of the lines x = 0, 0 y h, and x = 1, 0 y h. This is illustrated in Fig. 6. The length of this path is l 1 = m n d 1 (q i ) + h, i=1 where the notation d 1 (q i ) denotes the shortest distance of point q i from the nearest of the m + 1 lines. The second path is constructed similarly using the m lines y = h/2m,3h/2m,...,(2m 1)h/2m. This path also commences on y = h, passes through the aboe m lines (isiting the points wheneer they are at the shortest distance from these m lines) and ends on y = 0. The length of this path is n l 2 = (m + 2) + 2 d 2 (q i ) + h, i=1 where the notation d 2 (q i ) denotes the shortest distance of point q i from the nearest of the new m lines. Obsere that d 1 (q i ) + d 2 (q i ) = h/2m. Hence, l 1 + l 2 = 2m h + hn/m. Now choose m to be the integer nearest to hn/2, so that

8 n = 2(m + θ) 2 /h, where θ 1. Thus, l 1 + l 2 = 2m h + 2(m + θ) 2 /m = 4(m + θ) + 2h θ 2 /m 2 2hn + 2h + 5. Thus, at least one of the two paths must hae length upper bounded by 2hn + h + 5/2. (0, h) q 1 q 2 q 3 (0, 2h/m) (0, h/m) q 4 (0, 0) (1, 0) Fig. 6. Illustration of the proof of Theorem II.1. The dots indicate the locations of the points inside a rectangle of size 1 h. The first of the two paths considered in the proof through the points begins at (1, h) and follows the direction of the arrows, isiting a point wheneer it is within a distance of h/2m for a specific integer m from the solid horizontal lines.