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 serice demands arrie stochastically on a line segment. Upon arrial, the demands translate perpendicular to the line with a fixed speed. A ehicle, with speed greater than that of the demands, seeks to proide serice by reaching each mobile demand. In this paper we study a firstcome-first-sered FCFS) policy in which the serice ehicle seres demands in the order in which they arrie. hen the demand arrial rate is ery low, we show that the FCFS policy can be used to minimize the expected time, or the worstcase time, to serice a demand. e 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. hen the demands are much slower than the serice ehicle, the necessary and sufficient conditions become equal. e also show that in the limiting regime 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 minimizes the expected time to serice a demand. 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 distributed location on a line segment of length, see Fig.. 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). e 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/. e proposed a serice policy which relied on the computation of the translational minimum Hamiltonian path TMHP) through unsericed demands, and showed that for small the policy ensures stability for all arrial rates up to a constant factor of the necessary condition. In this paper we focus on the case when the arrial rate is low if is close to one we will see that this is a necessary condition for stability). For this case we propose a firstcome-first-sered FCFS) policy; such policies are common This material is based upon work supported in part by ARO-MURI Award 9NF-5--9, ONR Award N4-7--7 and by the Institute for Collaboratie Biotechnologies through the grant DAAD9-3-D-4 from the U.S. Army Research Office. 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 Fig.. The problem setup. The line segment is the generator of mobile demands. The dark circle denotes a demand and the square denotes a ehicle. in classical queuing theory [], [3]. In our proposed policy, the serice ehicle also seeks to optimize its position to respond to the arrial of new demands. Determining the optimal position is a coerage problem, and related works include geometric location problems such as [4], and [5], and robotic sensor coerage and deployment problems [6]. The contributions of this paper can be summarized as follows. e 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. e 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. e determine necessary and sufficient conditions on λ for the stability of the FCFS policy. As +, the necessary and sufficient conditions become equal. hen approaches one, we show that: i) for existence of a stabilizing policy, λ must tend to zero as / log ), ii) eery stabilizing policy must serice the demands in the order in which they arrie, and iii) the FCFS policy minimizes the expected time to serice a demand. hen compared to the TMHP-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.. This paper is organized as follows: the problem is formalized in Section II. The FCFS policy is introduced in Section II-B. In Section III we determine the optimal placement for minimizing the expected and worst-case trael time. 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. Due to space constraints, we limit the presentation of some proofs to a sketch. The complete proofs are presented in [7].
4 Necessary condition for stability of any policy Sufficient condition for stability of the TMHP based policy Sufficient condition for stability of the FCFS policy p = X,Y ) C = x,y + T) Arrial rate, ) q = x, y) 4 TMHP TMHP and FCFS FCFS.5 Fig. 3. Constant bearing control. The ehicle motion towards the point C := x, y + T) minimizes the time taken to reach the demand q. Arrial rate 8.. Arrial rate.98.99 Fig.. A summary of stability regions for the TMHP-based policy and the FCFS policy. Stable serice policies exist only for the region under the solid black cure. In the top and the bottom right figures, the red cure is due to Theorem V.. The solid black cure and the dashed blue cure in the top figure are described in []. In the asymptotic regime shown in the bottom right, the solid black cure is due to Theorem IV.3, and is different from the solid black cure in the top figure. In the asymptotic regime shown in the bottom left, the dashed blue cure is described in [], and is different than the one in the top figure. II. PROBLEM FORMULATION AND SERVICE POLICY e consider a single serice ehicle that seeks to serice mobile demands that arrie ia a spatio-temporal process on a segment with length 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. e 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. e define the enironment as E := [, ] 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 FINE) R, where FINE) is the set of finite subsets of E, 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, limsup i + E[D i ] < +, i.e., the steady state expected delay is finite. Equialently, the policy P is stable if under its action, lim sup 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 The ehicle uses the following motion, referred to as constant bearing control, to reach a moing demand. Proposition II. Constant bearing control, [8]) 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 )X x) Tp,q) := + Y y) Y y), minimizes the time taken by the ehicle to reach the demand. Constant bearing control is illustrated in Fig. 3. B. The First-Come-First-Sered FCFS) Policy e 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. e 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. 3 else 4 Moe using the constant bearing control to serice the furthest demand from the generator. 5 Repeat. Fig. 4 illustrates an instance of the FCFS policy. The first question is, how do we compute the optimal position X, Y )? This will be answered in the following section.
3 X, Y ).5, ) qi qi+ qi+ qi+3 Optimal Y position.5 Fig. 4. The FCFS policy. The ehicle serices the demands in the order of their arrial in the enironment, using the constant bearing control. 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). e determine placements that minimize the expected time and the worstcase time. A. Minimizing the Expected Time e 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 E[Tp,q)] to reach a demand generated at position q = x, ) from ehicle position p = X, Y ) is gien by ) )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 [, ] R >. ii) There exists a unique point p := /, Y ) R that minimizes E[Tp,q)]. Proof: [Sketch] Part i) follows from the fact that the Hessian of TX, Y ),, x)) with respect to X and Y, is positie semi-definite. Further, Tp, q) is strictly conex for all x /. But, letting p = /, Y ) and q =, x) we can write E[Tp,q)] = Tp,q)dx. ) x [,]\{/} The integrand is strictly conex for all x [, ] \ {/}, implying E[Tp,q)] is strictly conex on the line X = /, and the existence of a unique minimizer /, Y ), and part ii) is proen. 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 = /. Obtaining a closed.5...3.4.5.6.7.8.9 Demand elocity Fig. 5. 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 =. form expression for Y does not appear to be possible. E[Tp,q)] with X = /, yields E[Tp,q)] = Y b Y a ) ) log b a a 4Y, where a =, and b = + a /4Y. For each alue of and, this conex expression can be easily numerically minimized oer Y, to obtain Y. A plot of Y as a function of for = is shown in Fig. 5. 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. e now characterize the steady-state expected delay of the FCFS policy D FCFS, as λ tends to zero. Theorem III. FCFS optimality for low arrial rates) Fix any <. Then as λ +, D FCFS D, and the FCFS policy minimizes the expected time to serice a demand. Proof: e hae shown how to compute the position p := X, Y ) which minimizes E[Tp,q)]. 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 orst-case Time The expected time to serice a demand was the metric studied in the companion paper, and will be the metric of interest in 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 /, /).
Proof: [Sketch] The fact that X = / follows from symmetry. e then substitute this alue for X in Eq. II.), along with y =, to obtain the worst-case time expression as a function of Y. Performing a standard minimization of the worst-case time, we get Y = /. 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 ) = /, /), minimizes the worst-case time to serice a demand. IV. NECESSARY CONDITIONS FOR STABILITY In this section, we consider λ >, and determine necessary conditions on λ to ensure that the FCFS policy remains stable. To establish these conditions we utilize a standard result in queueing theory cf. []) which states that a necessary condition for the existence of a stabilizing policy is that λe[t], where E[T] is the expected time to serice a demand i.e., the trael time between demands). e also determine a necessary condition on λ for the stability of any policy as, and establish the optimality of the FCFS policy. e begin with the following. Proposition IV. Special case of equal speeds) For = there does not exist a stabilizing policy. Proof: [Sketch] If =, then a demand can be reached only if the serice ehicle is aboe the demand. Note that the only policy that ensures that a demand s y-coordinate neer exceeds that of the serice ehicle is the FCFS policy. The trael time between demand i and i + is gien by Tq i,q i+ ) = x + y y = ) x y + y, where x and y are the differences between their x- and y-coordinates respectiely. Taking expectation, and making use of the independence of x and y, we obtain E[Tq i,q i+ )] = +. This implies for eery λ >, λe[tq i,q i+ )] = +. This means that the necessary condition for stability, i.e., λe[tq i,q i+ )], is iolated. Thus, there does not exist a stabilizing policy. 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, for nec, λ 3 )), otherwise, + ) C nec log where C nec =.5+log) γ, where γ is the Euler constant; and nec is the solution to the equation + )C nec.5 log ) + log ) =, and is approximately equal to 4/5. Proof: [Sketch] The trael time between consecutie demands is gien by T = ) ) x + y y, where x and y are the differences in the x- and y- coordinates of the demands respectiely. Since T is conex in x and y, we apply Jensen s inequality, followed by substitution of the expressions for the expected alues of x and y, and obtain E[T] ) ) 9 + λ. λ Using the necessary condition for stability, λ 3. ) 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, applying Jensen s inequality, E[T y] ) /9 + y y, ) where E[ x] = /3. Now, the random ariable y is distributed exponentially with parameter λ/, un-conditioning Eq. ) on y we obtain that E[T] is lower bounded by λ ) + ) ) + y 9 y e λy/ dy. Using the software Maple R, this simplifies to π 3 λ [H ) 3 λ )] Y 3 λ ), where H ) is the st order Strue function and Y ) is st order Bessel function of the nd kind. A Taylor series expansion of H z) Y z) about z = yields H z) Y z) ) π z + C necz z logz), where C nec = / + log) γ.6. Thus, E[T] λ + ) + λ λ )) C nec log, 8 3 Using the necessary condition for stability, and simplifying using the fact that λ/3 <, we hae for stability, 3 λ + ) C nec log ) )), 3) when C nec > log /), i.e., when > nec, where nec is obtained when we set the RHS of Eq. ) equal to the RHS of Eq. 3), and is approximately equal to 4/5. Thus,
the necessary condition for stability is gien by Eq. ) when nec, and by Eq. 3) when > nec. The necessary condition states that in the limit as, λ goes to zero as / log ), which is slower than any polynomial in ). The following result shows that as, that the condition in Theorem IV. is necessary for eery policy. Theorem IV.3 Policy independent necessary condition) For the limiting regime as, eery stabilizing policy must sere the demands in the order in which they arrie and hence, 3 λ log ). Proof: [Sketch] Suppose there is a policy P that is not does not sere demands FCFS, but stabilizes the system with λ = B ) p, for some p >, and B >. As per P, suppose the ehicle seres demand i+j before demand i+. The time to trael to demand i + from any demand i + j, where j > is Tq i+j,q i+ ) 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 λtq i+j,q i+ ) B ) p ) p+/) +, as, making P unstable. 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. 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. The bound on λ follows from Theorem IV.. The following result is a consequence of Theorem IV.3 and the fact that the FCFS policy uses constant bearing control. Corollary IV.4 FCFS Optimality for high speed) In the limiting regime as, the FCFS policy minimizes the expected time to serice a demand. V. A SUFFICIENT CONDITION FOR FCFS STABILITY In this section, we derie a sufficient condition on the arrial rate that ensures stability for the FCFS policy. To establish this condition, we utilize a standard result in queueing theory cf. []) which states that a sufficient condition for the existence of a stabilizing policy is that λe[t] <, where E[T] is the expected time to serice a demand. Theorem V. Sufficient stability condition for FCFS) The FCFS policy is stable if 3 +, for suf, λ < )), otherwise, + ) C suf log where C suf = π/ log.5 3/ ), and suf is the solution to 3 )C suf log ) + log ) =, and is approximately equal to /3. Proof: e first upper bound the time taken T by the ehicle from position X, Y ), coinciding with a demand, to reach the next demand at x, y), using the inequality a + b a + b. Thus, X x Y y) T +, 4) Taking expectation, and using the sufficient condition for stability, λe[t] < λ < 3 +. 5) Eq. 4) gies a ery conseratie upper bound 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, and on following steps similar to those in the proof of Theorem IV., we obtain λ [H ) 6 π E[T] 6 λ )] Y 6 λ ). 6) e use upper bounds on the Strue and Bessel functions from [9] when is sufficiently large, i.e., when the argument of H ) and Y ) is small. It can be shown that H z) z, Y z) z π log z ), for z, 7) z where z := λ / 6). Substituting into Eq. 6), and upon simplification we obtain E[T] λ π log λ ) 3 3 log λ + ). 8) 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. 8) along with Eq. 5), we can obtain a lower bound on λ. Since λ < λ implies stability, a sufficient condition for stability is λ < + ) C log )), 9)
where C suf := π/ log.5 3/ ).6. To determine the alue of the speed beyond which this is a less conseratie condition than Eq. 5), we sole + ) C log )) = 3 +, which gies suf /3. It can be erified that for > suf, the argument of the Strue and Bessel functions is less than, and hence the bounds in Eq. 7) are alid.thus, a sufficient condition for FCFS stability is gien by Eq. 5) for suf and by Eq. 9) for > suf. Remark V. Limiting regimes) As +, the sufficient condition for FCFS stability becomes λ < 3/, 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 < log ), In comparison the necessary condition scales as λ 3 log ). 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. It should be noted that λ can conerge to zero extremely slowly as, and still satisfy the sufficient stability condition in Theorem V.. For example, with = 6, the FCFS policy stabilizes the system for an arrial rate of 3/5). VI. SIMULATIONS In this section, we present a numerical study to determine stability of the FCFS policy. e numerically determine the region of stability of the FCFS policy, and compare it with the theoretical results from the preious sections. For a gien alue of, λ), we begin with demands in the enironment and determine the ehicle s aerage y-coordinate at the end of the iteration. If it exceeds the y-coordinate at the beginning of the iteration, then that particular data point of, λ) is classified as being unstable; otherwise, it is stable. The results of this numerical experiment are presented in Figure 6. For the purpose of comparison, we oerlay the plots for the necessary and the sufficient conditions for FCFS stability, which were established in Theorems IV. and V. respectiely. e obsere that the numerically obtained stability boundary for the FCFS policy falls between the two theoretically established cures. VII. CONCLUSIONS AND FUTURE DIRECTIONS This two part paper has introduced a dynamic ehicle routing problem with moing demands. In this paper we studied the cases where the demands hae high speed and where the arrial rate of demands is low. e introduced a Arrial rate 4 3...3.4.5.6.7.8.9 Fig. 6. Numerically determined region of stability for the FCFS policy. A lightly shaded green-coloured) dot represents stability while a darkly shaded blue-coloured) dot represents instability. The lower red) cure is the sufficient stability condition in Theorem V.. The upper black) cure is the necessary stability condition in Theorem IV.. The enironment width is =. first-come-first-sered policy and gae necessary and sufficient conditions on the arrial rate for its stability. e also determined the optimal placement of the ehicle so as to minimize the worst-case, and the expected delay in sericing a demand. e showed that for fixed, as the arrial rate tends to zero, the FCFS policy minimizes the worst-case serice delay, and the expected serice delay. Finally we showed that as tends to one, FCFS minimizes the expected delay and that eery stabilizing policy must serice demands in the order in which arrie. e hae recently considered the case in which demands are approaching a deadline and the serice ehicle seeks to stop them []. Future directions include studying the case when demands are generated according to a nonuniform distribution on the generator, and the case of multiple ehicles. REFERENCES [] S. D. Bopardikar, S. L. Smith, F. Bullo, and J. P. 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