Dynamic Vehicle Routing with Moving Demands Part II: High speed demands or low arrival rates

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1 ACC 9, Submitted St. Louis, MO Dynamic Vehicle Routing with Moing Demands Part II: High speed demands or low arrial rates Stephen L. Smith Shaunak D. Bopardikar Francesco Bullo João P. Hespanha Abstract In the companion paper we introduced a ehicle routing problem in which demands arrie ia a temporal Poisson process, and uniformly distributed along a line segment. Upon arrial, the demands moe perpendicular to the line with a fixed speed. A serice ehicle, with speed greater than that of the demands, seeks to proide serice by reaching the location of each mobile demand. In this paper we study a first-comefirst-sered FCFS) policy in which the serice ehicle seres the demands in the order in which they arrie. When the demand arrial rate is ery low, we show that the FCFS policy can be used to minimize the expected time, or the worst-case time, spent by a demand before being sered. We determine necessary and sufficient conditions on the arrial rate of the demands as a function of the problem parameters) for the stability of the FCFS policy. When the demands are much slower than the serice ehicle the necessary and sufficient conditions become equal. We also show that in the limiting case when the demands moe nearly as fast as the serice ehicle; i) the demand arrial rate must tend to zero; ii) eery stabilizing policy must serice the demands in the order in which they arrie, and; iii) the FCFS policy is the optimal policy. I. INTRODUCTION In companion paper [] we introduced a ehicle routing problem in which demands arrie ia a temporal Poisson arrial process with rate λ at a uniformly random location on a line segment of length W. The demands moe in a fixed direction perpendicular to the line with fixed speed <. A serice ehicle, modeled as a unit speed firstorder integrator, seeks to sere these mobile demands by reaching each demands location. The goal is to determine conditions on the arrial rate λ, which ensure stability of the system i.e., ensure a finite expected time spent by a demand in the enironment). We refer the reader to [] for related work and motiation. In [] we showed that to ensure the existence of a stabilizing policy, we must hae λ 4/W. We proposed a serice policy which relied on the computation of the translational traeling salesperson path t- TSP) through unsericed demands, and showed that for small the policy ensures stability for all λ s up to a constant factor of the necessary condition. We now focus on the case when the arrial rate is low if is close to, but strictly smaller than one, we will see that this is a necessary condition for stability). For this case we propose a first-come-first-sered FCFS) policy; such policies are common in classical queuing theory [], [3]. In the regime where is fixed and λ tends to zero, the problem The authors are with the Center for Control, Dynamical Systems and Computation, Uniersity of California at Santa Barbara, Santa Barbara, CA 936, USA, {stephen,shaunak,bullo}@engineering.ucsb. edu, hespanha@ece.ucsb.edu becomes one of proiding optimal coerage. Related works include geometric location problems such as [4], and [5], where gien a set of static demand points in the plane, the goal is to find supply points that minimize a cost function of the distance from each demand to its nearest supply point. The authors in [6] study the problem of deploying robots into a region so as to proide optimal coerage of the region. The contributions of this paper can be summarized as follows. We study a first-come-first-sered FCFS) policy in which demands are sered in the order in which they arrie, and when the enironment contains no outstanding demands, the ehicle moes to a location which minimizes the expected or worst-case) trael time to a demand. We show that for fixed, as the demand arrial rate λ tends to zero, the FCFS policy is the optimal policy in terms of minimizing the expected or worst-case) delay between a demands arrial and its serice completion. Next, we determine necessary and sufficient conditions on λ for the stability of the FCFS policy. As +, the necessary and sufficient conditions become equal. When approaches one, we show that: i) for existence of a stabilizing policy, λ must conerge to zero as / log ), ii) eery stabilizing policy must serice the demands in the order in which they arrie, and iii) the FCFS policy is the optimal policy. When compared to the TSP-based policy introduced in companion paper [], the FCFS policy has a larger stability region when is large, but a smaller stability region when is small. This is summarized in Fig.. Arrial rate λ e f a TSP stable TSP and FCFS stable Stabilization impossible for any policy FCFS stable, ) Demand speed, ) Fig.. 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 b-c is due to Theorem IV., cures c-d and d-e are due to Theorem V.. For cures a-b and c-f, refer to []. b d c

2 This paper is organized as follows: the problem is formalized in Section II. The FCFS policy is introduced in Section II-B. We determine the optimal placement to minimize the expected time in Section III-A, and the worst-case time in Section III-B. In Section IV we determine a necessary condition for stability as tends to one, and in Section V we determine a sufficient condition for the stability of the FCFS policy. In Section VI, we present simulation results for the FCFS and its comparison with the TSP-based policy. II. PROBLEM FORMULATION AND SERVICE POLICY 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 >, and moe with constant speed < along the positie y-axis. 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. We define the enironment as E := [,W] R R, and let pt) = [Xt),Y t)] T E denote the position of the serice ehicle at time t. Let Qt) E denote the set of all demand locations at time t, and nt) the cardinality of Qt). 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 E R, assigning a commanded elocity to the serice ehicle as a function of the current state of the system: ṗt) = Ppt), Qt)). 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 i + E[D i ] < +, i.e., the steady state expected delay is finite. Equialently, the policy P is stable if under its action, lim E[nt)] < +, 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. A. Constant Bearing Control In this paper we will use the following result on catching a demand in minimum time. Definition II. Constant bearing control) Gien the locations p := X,Y ) E and q := x,y) E at time t of the serice ehicle and a demand, respectiely, with the demand moing in the positie y-direction with constant speed, the motion of the ehicle towards the point x,y + T), where )X x) Tp,q) := + Y y) Y y), ) with unit speed is defined as the constant bearing control. Constant bearing control is illustrated in Fig.., ) p = X,Y ) W C = x, y + T) q = x, y) Fig.. Constant bearing control. The ehicle moes towards the point C := x, y + T), where x, y, and T are as per Definition II., to reach the demand. The following result on constant bearing control is established in [7]. Proposition II. Minimum time control, [7]) The constant bearing control is the minimum time control for the ehicle to reach the demand. B. The First-Come-First-Sered FCFS) Policy We are now ready introduce the FCFS policy, which will be the focus of this paper. In this policy the serice ehicle uses constant bearing control and serices the demands in the order in which they arrie. If the enironment contains no demands, the ehicle moes to the location X,Y ) which minimizes the expected, or worst-case, time to catch the next demand to arrie. We can state this policy as follows. The FCFS policy Assumes: Gien the optimal location X,Y ) E. if no unsericed demands in E then Moe toward X,Y ) until the next demand arries. else Moe using the constant bearing control to serice the furthest demand from the generator Repeat. Fig. 3 illustrates an instance of the FCFS policy. The, ) qi qi+ X, Y ) Fig. 3. The FCFS policy. The ehicle serices the demands in the order of their arrial in the enironment, using the constant bearing control. W qi+ qi+3

3 first question is, how do we compute the optimal position X,Y )? This will be answered in the following section. III. OPTIMAL VEHICLE PLACEMENT In this section we study the FCFS policy when < is fixed and λ +. In this regime stability is not an issue, as demands arrie ery rarely, and the problem becomes one of optimally placing the serice ehicle ie., determining X,Y ) in the statement of the FCFS policy). We determine placements that minimize the expected time and the worstcase time. A. Minimizing the Expected Time We seek to place the ehicle at location that minimizes the expected time to serice a demand once it appears on the generator. Demands appear at uniformly random positions on the generator and the ehicle uses the constant bearing control to reach the demand. Thus, the expected time to reach a demand generated at position q = x, ) from ehicle position p = X,Y ) is gien by E[Tp,q)] = W ) W )X x) + Y Y ) dx. The following lemma characterizes the way in which this expectation aries with the position p. Lemma III. Properties of the expected time) i) The expected time E[Tp, q)] is conex in p, for all p [,W] R >. ii) There exists a unique point p := W/,Y ) R that minimizes E[Tp,q)]. Proof: For part i), it suffices to show that the integrand in Eq. ), Tp,q) is conex. To do this we compute the Hessian of TX,Y ),,x)) with respect to X and Y. Thus, for Y >, [ ] [ ] Y Y X x) T T Y X x) X x) X T Y X X Y T Y = ) 3/. )X x) + Y The Hessian is positie semi-definite, which implies that Tp,q) is conex in p for each q =,x). For part ii), obsere that since demands are uniformly randomly generated on the interal [,W], X has a unique minimum of W/. If we can show that E[Tp,q)] is strictly conex in Y when X = W/, then we hae proed part ii). From the T/ Y term of the Hessian we see that Tp,q) is strictly conex for all x W/. But, letting p = W/,Y ) and q =,x) we can write E[Tp,q)] = W ) x [,W]\{W/} Tp,q)dx. The integrand is strictly conex for all x [,W] \ {W/}, implying E[Tp,q)] is strictly conex on the line X = W/, and the existence of a unique minimizer W/,Y ). ) Optimal Y position Demand elocity Fig. 4. The Y position of the serice ehicle which minimizes the expected distance to a demand, as a function of. In this plot the generator has length W =. Lemma III. tells us that there exists a unique point p := X,Y ) which minimizes the expected trael time. In addition, we know that X = W/. Obtaining a closed form expression for Y does not appear to be possible. Computing the integral in Eq. ), with X = W/, one can obtain E[Tp,q)] = Y + aw a 4Y Y log aw + aw 4Y ) ) aw 4Y, where a =. For each alue of and W, this conex expression can be easily numerically minimized oer Y, to obtain Y. A plot of Y as a function of for W = is shown in Fig. 4. For the optimal position p, the expected delay between a demand s arrial and its serice completion is D := E[Tp,,x))]. Thus, a lower bound on the steady-state expected delay of any policy is D. We now characterize the steady-state expected delay of the FCFS policy D FCFS, as λ tends to zero. Theorem III. FCFS optimality) Fix any <. Then as λ +, D FCFS D, and the FCFS policy minimizes the expected time to serice a demand. Proof: We hae shown how to compute the position p := X,Y ) which minimizes Eq. ). Thus, if the ehicle is located at p, then the expected time to serice the demand is minimized. But, as λ +, the probability that demand i+ arries before the ehicle completes serice of demand i and returns to p tends to zero. Thus, the FCFS policy is optimal as λ +. B. Minimizing the Worst-Case Time In the preious section we looked at the expected time to serice a demand. This was the metric studied in the companion paper, and will be the metric of interest in

4 Section V when we study the FCFS policy for λ >. Howeer, another metric that can be used to determine X,Y ) is the worst-case time to serice a demand. Lemma III.3 Optimal placement for worst-case) The location X,Y ) that minimizes the worst-case time to serice the demand is W/, W/). At this location, in the worst-case, the ehicle moes a distance of W/ horizontally. Proof: Assume that X > W/ or X < W/). Then the worst-case time is achieed when a target appears at the location,) resp. W,)). One can then moe the ehicle parallel to the x-axis such that X = W/ and thus decrease the time taken to serice the same target as compared to its preious position. This is a contradiction. Hence, X = W/. Applying the constant bearing control for the ehicle, we obtain the worst case system time as T w Y ) = )W /4 + Y Y Since T w is solely a function of Y, to minimize T w, we must hae dt w Y ) Y dy = Y )W /4 + Y ) =. Simplifying, we get Y = W/. Using an argument identical to that in the proof of Theorem III. we hae the following: For fixed <, and as λ +, the FCFS policy, with X,Y ) = W/,W/), minimizes the worst-case time to serice a demand. IV. NECESSARY CONDITIONS FOR STABILITY In the preious section, we studied the case of fixed and low λ +. In this section we look at the problem when λ >, and determine necessary conditions on the magnitude of λ that ensure the FCFS policy remains stable. We also determine a necessary condition on λ for the stability of any policy as, and establish the optimality of the FCFS policy. We begin with the following. Lemma IV. Special case of equal speeds) For = there does not exist a stabilizing policy. Proof: When =, gien a ehicle location p := X,Y ) and a demand location with initial location q := x,y), the minimum time T in which the ehicle can reach the demand is gien by Tp,q) = X x) + Y y) Y y) ). if Y > y, 3) and is undefined if Y y. Thus, a demand can only be reached if the ehicle is aboe the demand. From Eq. 3) we see that a necessary stability condition is that a demand s y- coordinate neer exceeds that of the serice ehicle. The only policy that can ensure this is the FCFS policy. Thus, we proe the result by computing the expected time to trael between demands using the FCFS policy, and show that for eery λ >, more than one demand arries during this trael time. To do this, assume there are many outstanding demands below the serice ehicle, and none aboe. Suppose the serice ehicle completed the serice of demand i at time t i and position x i t i ),y i t i )). Let us compute the expected time to reach demand i +, with location x i+ t i ),y i+ t i )). Since arrials are Poisson it follows that y i t i ) > y i+ t i ). To simplify notation we define x = x i t i ) x i+ t i ) and y = y i t i ) y i+ t i ). Then, from Eq. 3) Tq i,q i+ ) = x + y = ) x y y + y. Taking expectation and noting that x and y are independent E[Tq i,q i+ )] = E [ x ] [ ] ) E y + E[ y]. Now, E[ y] = /λ, E [ x ] is a positie constant independent of λ and [ ] + E y = y λe λy dy = +. y= Thus E[Tq i,q i+ )] = +, and for eery λ >, λe[tq i,q i+ )] = +, implying that an infinite number of demands arrie in the time required to serice one. Next we look at the FCFS policy and gie a necessary condition for its stability. Theorem IV. Necessary stability condition for FCFS) A necessary condition for the stability of the FCFS policy is 3 W, for 4 5, λ 3 )), o.w., W + ) A log where A.6. Proof: Suppose the serice ehicle completed the serice of demand i at time t i at position x i t i ),y i t i )) and demand i + is located at x i+ t i ),y i+ t i )). Also define x = x i t i ) x i+ t i ) and y = y i t i ) y i+ t i ). For <, the trael time between demands is gien by T = ) ) x + y y. Obsere that the function T is conex in x and y. So by applying Jensen s inequality, we obtain E[T] ) )E[ x]) + E[ y]) E[ y]. Substituting the expressions for the expected alues, E[T] ) W ) 9 + λ. λ

5 From the necessary condition for stability, cf. [8]), we must hae )W λe[t] λ ) + 9 λ. λ On simplifying, we obtain λ 3 W. 4) This proides a good necessary condition for low, but we will be able to obtain a much better necessary condition for large. Since T is conex in x, we can apply Jensen s inequality to write E[T y] )W /9 + y y, 5) where E[ x] = W/3. Now, the random ariable y is distributed exponentially with parameter /λ and probability density function fy) = λ e λy/. Un-conditioning Eq. 5) on y we obtain E[T] = λ ) + + E[T y]fy)dy ) )W + y 9 y e λy/ dy. 6) The right hand side can be ealuated using the software Maple R and equals πw 3 λw [H ) 3 λw )] Y 3 λ ), where H ) is the st order Strue function and Y ) is st order Bessel function of the nd kind. We can perform a Taylor series expansion of the function H z) Y z) about z = to obtain H z) Y z) ) + Az z logz), π z where A = / + log) γ.6. Using the aboe expression, Eq. 6) can be written as E[T] λ + ) + λw λw )) A log, 8 3 where we hae used the fact that λ ) λ ) = λ + ). To obtain a stability condition on λ we wish to remoe λ from the log term. To do this, note that from Eq. 4) we hae ) λw/3 <, and thus E[T] λ + ) + λw 8 λ + ) + λw 8 A log Wλ 3 log W ), 3 )) A log For stability we require that λe[t], from which we see that a necessary condition for stability is )) λ W A log 8 + = +. Soling for λ when A > log /) we obtain that 3 λ W + ) A log )), 7) The condition A > log /), implies that the aboe bound holds for all < / + e A /. We now hae two bounds; Eq. 4) which holds for all <, and Eq. 7) which holds for > /. The final step is to determine the alues of for which each bound is actie. To do this, we set the right-hand side of Eq. 4) equal to the RHS of Eq. 7) and sole for to obtain.8. Thus, the necessary condition for stability is gien by Eq. 4) when.8, and by Eq. 7) when >.8. The preious theorem shows that although λ must go to zero as, it can go ery slowly to. In fact, the necessary condition states that λ goes to zero as log ). This goes to zero more slowly than any polynomial in ). Finally, we proe that the FCFS is the best policy as. Theorem IV.3 Optimality of FCFS) For the limiting case as ; i) eery stabilizing policy must sere the demands in the order in which they arrie; ii) the stability condition in Theorem IV. is necessary for all policies; and iii) no policy can proide a lower expected delay than the FCFS policy. Proof: We begin by proing Part i). Parts ii) and iii) are direct consequences. Suppose there is a policy P that is not does not sere demands FCFS, but can stabilize the system with λ = B ) p, for some p >, and B >. Let t i be the first instant at which policy P deiates from FCFS. Then, the demand sered

6 immediately after i is demand i+k for some k >. When the ehicle reaches demand i+k at time t i+, demand i+ has moed aboe the ehicle. To ensure stability, demand i + must eentually be sered. The time to trael to demand i+ from any demand i + j, where j > is Tq i+j,q i+ ) = x ) + y ) + y y + y = y, where x and y are now the minimum of the x and y distances from q i+j to the q i+. The random ariable y is Erlang distributed with shape j and rate λ, implying P[ y c] e λc/, for each c >. Now, since λ = B ) p as, almost surely y > ) / p. Thus Tq i+j,q i+ ) ) p+/), almost surely as. Thus, the expected number of demands that arrie during Tq i+j,q i+ ) is λtq i+j,q i+ ) B ) p ) p+/) B ) / +, as. This implies that almost surely the policy P becomes unstable when it deiates from FCFS. Thus, any deiation must occur with probability zero as. Thus, a necessary condition for a policy to stabilize with λ = B ) p, is that as, the policy must sere demands in the order in which they arrie. But this holds for eery p, and by letting p go to infinity, B ) p conerges to zero for all,]. Thus, a non-fcfs policy cannot stabilize the system no matter how quickly λ + as. Hence, as, eery stabilizing policy must sere the demands in the order in which they arrie. To see parts ii) and iii), notice that the definition of the FCFS policy is that it uses the minimum time control i.e., constant bearing control) to moe between demands, thus the necessary stability condition in Theorem IV. holds for all policies as. In addition, the FCFS spends the minimum amount of time to trael between demands and thus minimizes the expected delay. V. A SUFFICIENT CONDITION FOR FCFS STABILITY In Section IV we determined a necessary condition for stability of the FCFS policy. In this section, we will derie the following sufficient condition on the arrial rate that ensures stability for the FCFS policy. Theorem V. Sufficient stability condition for FCFS) The FCFS policy is stable if 3 W +, for 3, λ < W + ) C log )), o.w., where C.6. Proof: We begin with the expression for the time taken for the ehicle from the position p, coinciding with a demand, to reach the next demand at q using the constant bearing control cf. Definition II.). Thus, )X x) Tp,q) = + Y y) Y y) X x Y y) +, 8) where we used the inequality a + b a + b. Taking expectation, E[T] W 3 + λ ), since the demands are distributed uniformly in the x- direction and Poisson in the y-direction. A sufficient condition for stability is cf. [8]) λe[t] < λ < 3 W +. 9) The upper bound on T gien by Eq. 8) is ery conseratie except for the case when is ery small. Alternatiely, taking expected alue of T conditioned on y, and applying Jensen s inequality to the square-root part, we obtain E[T y] )W /6 + y y ), since E [ x ] = W /6. Following steps which are similar to those between Eq. 5) and Eq. 6), we obtain πw E[T] 6 λw [H ) 6 λw )] Y 6 λ ), ) where H ) is the st order Strue function and Y ) is st order Bessel function of the nd kind. In [9], polynomial approximations hae been proided for the Strue and Bessel functions in the interals [,3] and [3,+ ). We seek an upper bound for the right-hand side of ) when is sufficiently large, i.e., when the argument of H ) and Y ) is small. Thus, from [9] H z) z, Y z) z π log z ), for z 3, z where z := λw / 6). Substituting into Eq. ), we obtain [ πw E[T] 6 λw π λw λw log λw )] 6 6 λ ),

7 which yields E[T] λw π log λw 3 log ) 3 λ + ). ) Now, let λ be the least upper bound on λ for which the FCFS policy is unstable, i.e., for eery λ < λ, the FCFS policy is stable. To obtain λ, we need to sole λ E[T] =. Using Eq. ), we can obtain a lower bound on λ by simplifying λ W π log λ W 3 ) 3 ) log + ) +. From the condition gien by Eq. 9), the second term in the parentheses satisfies λ W > 3 +. Thus, we obtain, λ W + ) C log )), where the constant C = π/ log.5 3/ ).6. Since λ < λ implies stability, a sufficient condition for stability is λ < W + ) C log )). ) To determine the alue of the speed beyond which this is a less conseratie condition than Eq. 9), we sole W + ) C log )) = 3 W +, which gies /3. For >, one can erify that the numerical alue of the argument of the Strue and Bessel functions is less than 3, and so the approximation used in this analysis is alid. Thus, a sufficient condition for stability is gien by Eq. 9) for /3, and by Eq. ) for > /3. Remark V. Limiting regimes) As +, the sufficient condition for FCFS stability becomes λ < 3/W, which is exactly equal to the necessary condition gien by part ii) of Theorem IV.. Thus, the condition for stability is asymptotically tight in this limiting regime. As, the sufficient condition for FCFS stability becomes 6 λ < W log ), In comparison the necessary condition scales as λ 3 W log ). Arrial rate λ Necessary condition Sufficient condition Demand speed Fig. 5. The necessary and sufficient conditions for the stability of the FCFS policy. Thus, the necessary and sufficient conditions for the stability of the FCFS policy and by Theorem IV.3, for any policy) differ by a factor 3. Fig. 5 shows a comparison of the necessary and sufficient stability conditions for the FCFS policy. It should be noted that λ can conerge to zero extremely slowly as tends to one, and still satisfy the sufficient stability condition in Theorem V.. For example, with = , the FCFS policy can stabilize the system for an arrial rate of 3/5W). VI. SIMULATIONS In this section we present results of numerical experiments of the FCFS policy. 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 demands, large enough to ensure steady state was reached. We obtained the steady state delay by computing the mean of the delay in the last iterations. This was repeated 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 four different alues of the speed. The ariation of the expected steady-state delay with the generation rate is presented in Fig. 6 for different alues of the demand speed. The system is obsered to be stable for generation rates up to the sufficient condition for stability, which was theoretically established in Section V. We also compared the steady-state expected delay of the FCFS policy to the TSP-based policy proposed in companion paper [], in the region where both policies are stable c.f., Fig. ). The comparison was performed for two alues of speed, =.5 and =.5, arying λ from zero up to alues close to the stability limit of FCFS. The results are shown in Fig. 7. We obsere that in this region, the FCFS performs significantly better in terms of the steadystate expected delay than the TSP-based policy. VII. CONCLUSIONS This two part paper has introduced a dynamic ehicle routing problem with moing demands. In this paper we

8 Expected serice time =. =.5 =.8 = Generation rate λ Fig. 6. Simulation results for FCFS policy: ariation of the steady state expected delay with the arrial rate for four different alues of the demand speeds. The width of the enironment is W = 5. Expected serice time TSP based FCFS = Generation rate λ ACKNOWLEDGMENTS This material is based upon work supported in part by ARO-MURI Award W9NF-5--9, ONR Award N and by the Institute for Collaboratie Biotechnologies through the grant DAAD9-3-D-4 from the U.S. Army Research Office. REFERENCES [] S. D. Bopardikar, S. L. Smith, F. Bullo, and J. P. Hespanha, Dynamic ehicle routing with moing demands Part I: Low speed demands and high arrial rates, in American Control Conference, St. Louis, MO, June 9, submitted. [] L. Kleinrock, Queueing Systems. Volume I: Theory. New York: John Wiley, 975. [3], Queueing Systems. Volume II: Computer Applications. New York: John Wiley, 976. [4] N. Megiddo and K. J. Supowit, On the complexity of some common geometric location problems, SIAM Journal on Computing, ol. 3, no., pp. 8 96, 984. [5] E. Zemel, Probabilistic analysis of geometric location problems, SIAM Journal on Algebraic and Discrete Methods, ol. 6, no., pp. 89, 984. [6] J. Cortés, S. Martínez, T. Karatas, and F. Bullo, Coerage control for mobile sensing networks, IEEE Transactions on Robotics and Automation, ol., no., pp , 4. [7] R. Isaacs, Differential games. New York: John Wiley, 965. [8] D. J. Bertsimas and G. J. an Ryzin, A stochastic and dynamic ehicle routing problem in the Euclidean plane, Operations Research, ol. 39, pp. 6 65, 99. [9] J. N. Newman, Approximations for the Bessel and Strue functions, Mathematics of Computation, ol. 43, no. 68, pp , =.5 Expected serice time TSP based FCFS Generation rate λ Fig. 7. Comparison of the steady state expected delay using the FCFS and the TSP-based policy, for low alues of arrial rate λ. Each data point is obtained after running the policy 3 times. The width of the enironment is W = 5. studied the cases where the demands hae high speed and where the arrial rate of demands is low. We introduced a first-come-first-sered policy and gae necessary and sufficient conditions on the arrial rate for its stability. We also determined the optimal placement of the ehicle so as to minimize the worst-case, and the expected delay in sericing a demand. We then showed that for fixed, as the arrial rate tends to zero, no policy can perform better than FCFS in terms of minimizing the worst-case serice delay, and the expected serice delay. Finally we showed that as tends to one, FCFS is the optimal policy, as eery stabilizing policy must serice demands in the order in which arrie. For future work we will extend our results to the case when demands are generated according to a nonuniform distribution on the generator, and the case of multiple ehicles. This extension has been completed for the placement problem.