Dynamic Vehicle Routing for Translating Demands:
Stability Analysis and Receding-Horizon Policies
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Citation
Bopardikar, S.D. et al. “Dynamic Vehicle Routing for Translating
Demands: Stability Analysis and Receding-Horizon Policies.”
Automatic Control, IEEE Transactions On 55.11 (2010) : 25542569. © 2011 IEEE.
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http://dx.doi.org/10.1109/tac.2010.2049278
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Institute of Electrical and Electronics Engineers
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 11, NOVEMBER 2010
Dynamic Vehicle Routing for Translating Demands:
Stability Analysis and Receding-Horizon Policies
Shaunak D. Bopardikar, Member, IEEE, Stephen L. Smith, Member, IEEE, Francesco Bullo, Fellow, IEEE, and
João P. Hespanha, Fellow, IEEE
Abstract—We introduce a problem in which demands arrive
stochastically on a line segment, and upon arrival, move with a
fixed velocity perpendicular to the segment. We design a receding
horizon service policy for a vehicle with speed greater than that of
the demands, based on the translational minimum Hamiltonian
path (TMHP). We consider Poisson demand arrivals, uniformly
distributed along the segment. For a fixed segment width and
fixed vehicle speed, the problem is governed by two parameters;
the demand speed and the arrival rate. We establish a necessary
condition on the arrival rate in terms of the demand speed for the
existence of any stabilizing policy. We derive a sufficient condition
on the arrival rate in terms of the demand speed that ensures
stability of the TMHP-based policy. When the demand speed
tends to the vehicle speed, every stabilizing policy must service
the demands in the first-come-first-served (FCFS) order; and
the TMHP-based policy becomes equivalent to the FCFS policy
which minimizes the expected time before a demand is serviced.
When the demand speed tends to zero, the sufficient condition on
the arrival rate for stability of the TMHP-based policy is within
a constant factor of the necessary condition for stability of any
policy. Finally, when the arrival rate tends to zero for a fixed
demand speed, the TMHP-based policy becomes equivalent to the
FCFS policy which minimizes the expected time before a demand
is serviced. We numerically validate our analysis and empirically
characterize the region in the parameter space for which the
TMHP-based policy is stable.
Index Terms—Autonomous vehicles, dynamic vehicle routing,
minimum Hamiltonian path, queueing theory.
I. INTRODUCTION
EHICLE routing problems are concerned with planning
optimal vehicle routes for providing service to a given set
of customers. The routes are planned with complete information of the customers, and thus the optimization is static, and
typically combinatorial [3]. In contrast, dynamic vehicle routing
(DVR) considers scenarios in which not all customer information is known a priori, and thus routes must be re-planned as
V
Manuscript received March 17, 2009; revised November 17, 2009; accepted
February 27, 2010. Date of publication April 29, 2010; date of current version November 03, 2010. This work was supported in part by ARO-MURI
Award W911NF-05-1-0219, ONR Award N00014-07-1-0721 and by the Institute for Collaborative Biotechnologies through grant DAAD19-03-D-0004 from
the U.S. Army Research Office. Recommended by Associate Editor I. Paschalidis.
S. D. Bopardikar, F. Bullo, and J. P. Hespanha are with the Center for Control,
Dynamical Systems and Computation, University of California at Santa Barbara, Santa Barbara, CA 93106 USA (e-mail: shaunak@engineering.ucsb.edu;
bullo@engineering.ucsb.edu; hespanha@ece.ucsb.edu).
S. L. Smith is with the Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge MA 02139 USA
(e-mail: slsmith@mit.edu).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TAC.2010.2049278
new customer information becomes available. DVR problems
naturally occur in scenarios where autonomous vehicles are deployed in complex and uncertain environments. Examples include search and reconnaissance missions, and environmental
monitoring. An early DVR problem is the dynamic traveling
repairperson problem (DTRP) [4], in which customers, or demands arrive sequentially in a region and a service vehicle seeks
to serve them by reaching each demand location. In this paper,
we introduce a dynamic vehicle routing problem in which the
demands move with a specified velocity upon arrival, and we design a novel receding horizon control policy for a single vehicle
to service them. This problem has applications in areas such as
surveillance and perimeter defense, wherein the demands could
be visualized as moving targets trying to cross a region under
surveillance by an Unmanned Air Vehicle [5], [6]. Another application is in the automation industry where the demands are
objects that arrive continuously on a conveyor belt and a robotic
arm performs a pick-and-place operation on them [7].
A. Contributions
We introduce a dynamic vehicle routing problem in which
demands arrive via a stochastic process on a line segment of
fixed length, and upon arrival, translate with a fixed velocity
perpendicular to the segment. A service vehicle, modeled as a
first-order integrator having speed greater than that of the demands, seeks to serve these mobile demands. The goal is to design stable service policies for the vehicle, i.e., the expected time
spent by a demand in the environment is finite. We propose a
novel receding horizon control policy for the vehicle that services the translating demands as per a translational minimum
Hamiltonian path (TMHP).
In this paper, we analyze the problem when the demands are
uniformly distributed along the segment and the demand arrival
of the segprocess is Poisson with rate . For a fixed length
ment and the vehicle speed normalized to unity, the problem is
governed by two parameters; the demand speed and the arrival
rate . Our results are as follows. First, we derive a necessary
condition on in terms of for the existence of a stable service
policy. Second, we analyze our novel TMHP-based policy and
derive a sufficient condition for in terms of that ensures stability of the policy. With respect to stability of the problem, we
identify two asymptotic regimes: a) High speed regime: when
the demands move as fast as the vehicle, i.e.,
(and
therefore for stability,
); and b) High arrival regime:
when the arrival rate tends to infinity, i.e.,
(and therefore for stability,
). In the high speed regime, we show
that: i) for existence of a stabilizing policy, must converge to
zero as
, ii) every stabilizing policy must service the demands in the first-come-first-served (FCFS) order,
0018-9286/$26.00 © 2010 IEEE
BOPARDIKAR et al.: DYNAMIC VEHICLE ROUTING FOR TRANSLATING DEMANDS
Fig. 1. Summary of stability regions for the TMHP-based policy. Stable service
policies exist only for the region under the solid black curve. In the top figure,
the solid black curve is due to part (i) of Theorem III.1 and the dashed blue
curve is due to part (i) of Theorem III.2. In the asymptotic regime shown in the
bottom left, the dashed blue curve is described in part (ii) of Theorem III.2, and
is different than the one in the top figure. In the asymptotic regime shown in the
bottom right, the solid black curve is due to part (ii) of Theorem III.1, and is
different from the solid black curve in the top figure.
and iii) the TMHP-based policy becomes equivalent to the FCFS
policy which we establish is optimal in terms of minimizing the
expected time to service a demand. In the high arrival regime,
we show that the sufficient condition on for the stability of the
TMHP-based policy is within a constant factor of the necessary
condition on for stability of any policy. Third, we identify another asymptotic regime, termed as the low arrival regime, in
which the arrival rate
for a fixed demand speed. In this
low arrival regime, we establish that the TMHP-based policy
becomes equivalent to the FCFS policy which we establish is
optimal in terms of minimizing the expected time to service a
demand. Fourth, for the analysis of the TMHP-based policy, we
study the classic FCFS policy in which demands are served in
the order in which they arrive. We determine necessary and sufficient conditions on for the stability of the FCFS policy. Fifth
and finally, we validate our analysis with extensive simulations
and provide an empirically accurate characterization of the region in the parameter space of demand speed and arrival rate
for which the TMHP-based policy is stable. Our numerical results show that the theoretically established sufficient stability
condition for the TMHP-based policy in the high arrival asymptotic regime serves as a very good approximation to the stability
boundary, for nearly the entire range of demand speeds.
A plot of the theoretically established necessary and sufficient
conditions for stability in the - parameter space is shown in
Fig. 1. The bottom figures are for the asymptotic regimes of
, and
, respectively.
B. Related Work
Dynamic vehicle routing (DVR) is an area of research that
integrates fundamentals of combinatorics, probability and
queueing theories. One of the early versions of a DVR problem
is the Dynamic Traveling Repairperson Problem [4] in which
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the goal is to minimize the expected time spent by each demand
before being served. In [4], the authors propose a policy that
is optimal in the case of low arrival rate, and several policies
within a constant factor of the optimal in the case of high
arrival rate. In [8], they consider multiple vehicles, and vehicles
with finite service capacity. In [9], a single policy is proposed
which is optimal for the case of low arrival rate and performs
within a constant factor of the best known policy for the case
of high arrival rate. Recently, there has been an upswing in
versions of DVR such as in [10] where different classes of
demands have been considered; and in [11] which addresses the
case of demand impatience. DVR problems addressing motion
constraints on the vehicle have been presented in [12], while
limited sensing range for the vehicle has been considered in
[13]. Problems involving multiple vehicles and with minimal
communication have been considered in [14]. In [15], adaptive and decentralized policies are developed for the multiple
service vehicle versions, and in [16], a dynamic team-forming
variation of the Dynamic Traveling Repairperson problem is
addressed. Related dynamic problems include [17], wherein a
receding horizon control has been proposed for multiple vehicles to visit target points in uncertain environments; and [18],
wherein pickup and delivery problems have been considered.
Another related area of research is the version of the Euclidean Traveling Salesperson Problem (ETSP) with moving
points or objects. The static version of the ETSP consists of determining the minimum length tour through a given set of static
points in a region [19]. Vehicle routing with objects moving on
straight lines was introduced in [7], in which a fixed number
of objects move in the negative direction with fixed speed,
and the motion of the service vehicle is constrained to be parallel to either the or the axis. For a version of this problem
wherein the vehicle has arbitrary motion, termed as the translational Traveling Salesperson Problem, a polynomial-time approximation scheme has been proposed in [20] to catch all objects in minimum time. Another variation of this problem with
object motion on piece-wise straight line paths, and with different but finite object speeds has been addressed in [21]. Other
variants of the ETSP in which the points are allowed to move in
different directions have been addressed in [22].
A third area of research related to this work is a geometric location problem such as [23], and [24], where given a set of static
demands, the goal is to find supply locations that minimize a cost
function of the distance from each demand to its nearest supply
location. In our problem, in the asymptotic regime of low arrival,
when the arrival rate tends to zero for a fixed demand speed
, the problem becomes one of providing optimal coverage. In
this regime, the demands are served in a first-come-first-served
order; such policies are common in classical queueing theory
[25]. Another related work is [26], in which the authors study
the problem of deploying robots into a region so as to provide
optimal coverage.
Other relevant literature include [27] in which multiple mobile targets are to be kept under surveillance by multiple mobile
sensor agents. The authors in [28] propose mixed-integer-linearprogram approach to assigning multiple agents to time-dependent cooperative tasks such as target tracking.
In an early version of this work [1], [2], we analyzed a preliminary version of the TMHP-based policy and the FCFS policy
respectively. In this paper, we present and analyze a unified
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 11, NOVEMBER 2010
minimum Hamiltonian path (EMHP) is a Euclidean Hamiltonian path that has minimum length. In this paper, we also
consider the problem of determining a constrained EMHP
which starts at a specified start point, visits a set of points and
terminates at a specified end point.
More specifically, the EMHP problem is as follows.
Fig. 2. Constant bearing control. The vehicle moves towards the point C :=
x; y + vT ), where x, y , v and T are as per Definition II.1, to reach the demand.
(
policy for the problem. We also present new simulation results
to numerically determine the region of stability for the policy
proposed in this paper. As in [17] and in [18], the problem addressed in this paper applies several control theoretic concepts to
a relevant robotic application. In particular, we design policies
or control laws in order to stabilize a stochastic and dynamic
system. The solution that we propose is a version of receding
horizon control, in which one computes the optimal solution
based on all of the present information and then repeatedly recomputes as new information becomes available. Our stability
analysis relies on writing the closed-loop dynamical system as
a recursion in a state of the system, and then ensuring that this
state has bounded evolution.
C. Organization
This paper is organized as follows. A short review of results
on optimal motion and combinatorics is presented in Section II.
The problem formulation, the TMHP-based policy, and the main
results are presented in Section III. The FCFS policy is presented and analyzed in Section IV. Utilizing the results of Section IV, the main results are proven in Section V. Finally, simulation results are presented in Section VI.
Given static points placed in , determine the length
of the shortest path which visits each point exactly once.
An upper bound on the length of such a path for points in a
unit square was given by Few [30]. Here we extend Few’s bound
to the case of points in a rectangular region. For completeness,
we have included the proof in Appendix.
Lemma II.3 (EMHP Length): Given points in a
rectangle in the plane, where
, there exists a path that starts
from a unit length edge of the rectangle, passes through each of
the points exactly once, and terminates on the opposite unit
length edge, with length upper bounded by
Given a set
of points in , the ETSP is to determine
the shortest tour, i.e., a closed path that visits each point exactly
denote the length of the ETSP tour through
once. Let
. The following is a classic result from [31].
Theorem II.4 (Asymptotic ETSP Length, [31]): If a set of
points are distributed independently and uniformly in a compact
region of area , then there exists a constant
such that,
with probability one
(2)
II. PRELIMINARY RESULTS
In this section, we provide some useful background results.
A. Constant Bearing Control
The following control is used to reach a moving demand.
Definition II.1 (Constant Bearing Control): Given initial locations
and
of the vehicle
and a demand, respectively, with the demand moving in the positive -direction with constant speed
, the motion of
the service vehicle towards the point
, where
(1)
with unit speed is defined as the constant bearing control.
Shown in Fig. 2, the constant bearing control is known to
reach a demand in minimum time as is characterized in the following proposition.
Proposition II.2 (Minimum Time Control, [29]): The constant bearing control is the minimum time control for the service
vehicle to reach the moving demand.
B. Euclidean and Translational Minimum Hamiltonian Path
(EMHP/TMHP) Problems
Given a set of points in the plane, a Euclidean Hamiltonian
path is a path that visits each point exactly once. A Euclidean
The constant
has been estimated numerically as
, [32].
Next, we describe the TMHP problem which was proposed
and solved in [20]. This problem is posed as follows.
Given initial coordinates;
of a start point,
of a set of points, and of a finish point, all
moving with the same constant speed and in the same direction, determine a path that starts at time zero from point ,
visits all points in the set exactly once and ends at the finish
point, and the length
of which is minimum.
We wish to determine the TMHP through points which translate in the positive direction. We also assume the speed of the
service vehicle to be normalized to unity, and hence consider the
speed of the points
. A solution for the TMHP problem
is the Convert-to-EMHP method:
i) For
, define the conversion map
by
ii) Compute the EMHP that starts at
, passes through
the set of points given by
and ends
at
.
iii) Move between any two demands using the constant
bearing control.
The following result is established.
BOPARDIKAR et al.: DYNAMIC VEHICLE ROUTING FOR TRANSLATING DEMANDS
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Fig. 4. An iteration of the TMHP-based policy. The vehicle shown as a square
serves all outstanding demands shown as black dots as per the TMHP that begins
at (
) and terminates at
which is the demand with the least -coordinate. The first figure shows a TMHP at the beginning of an iteration. The second
figure shows the vehicle servicing the demands through which the TMHP has
been computed while new demands arrive in the environment. The third figure
shows the vehicle repeating the policy for the set of new demands when it has
completed service of the demands present at the previous iteration.
X;Y
Fig. 3. Problem set-up. The thick line segment is the generator of mobile demands. A dark circle denotes a demand and the square denotes the vehicle.
Lemma II.5 (TMHP Length, [20]): Let the initial coordinates
and
, and the speed of the points
. The length of the TMHP is
where
denotes the
length of the EMHP with starting point
, passing through
, and ending at
.
the set of points
This lemma implies the following result: given a start point,
a set of points and an end point all of whom translate in the
positive vertical direction at speed
, the order of the
points followed by the optimal TMHP solution is the same as
the order of the points followed by the optimal EMHP solution
through a set of static locations equal to the locations of the
moving points at initial time converted via the map .
III. PROBLEM STATEMENT AND TMHP-BASED POLICY
In this section, we pose the dynamic vehicle routing problem
with translating demands and present the TMHP-based policy
along with the main results.
A. Problem Statement
We consider a single service vehicle that seeks to service
mobile demands that arrive via a spatio-temporal process on a
line segment with length
along the -axis, termed the generator. The vehicle is modeled as a first-order integrator with
speed upper bounded by one. The demands arrive uniformly distributed on the generator via a temporal Poisson process with intensity
, and move with constant speed
along the
positive -axis, as shown in Fig. 3. We assume that once the vehicle reaches a demand, the demand is served instantaneously.
The vehicle is assumed to have unlimited fuel and demand servicing capacity.
We define the environment as
,
and let
denote the position of the
service vehicle at time . Let
denote the set of all
demand locations at time , and
the cardinality of
.
Servicing of a demand
and removing it from the set
occurs when the service vehicle reaches the location of the demand. A static feedback control policy for the system is a map
, where
is the set of finite subsets of ,
q
y
assigning a commanded velocity to the service vehicle as a func.
tion of the current state of the system:
Let
denote the time that the th demand spends inside the set
, that is, the delay between the generation of the th demand
and the time it is serviced. The policy is stable if under its
action
that is, the steady state expected delay is finite. Equivalently, the
policy is stable if under its action
that is, if the vehicle is able to service demands at a rate that
is—on average—at least as fast as the rate at which new demands arrive. In what follows, our goal is to design stable control policies for the system.
To obtain further intuition into stability of a policy, consider
the - parameter space. In the asymptotic regime of high speed,
where
, the arrival rate must tend to zero for stability,
otherwise the service vehicle would have to move successively
further away from the generator in expected value, thus making
the system unstable. Similarly, stability in the asymptotic region of high arrival, where
implies that the demand
speed must tend to zero. Thus, our primary goals are: i) to
characterize regions in the - parameter space in which one
can never design any stable policy, and ii) design a novel policy
and determine its stability region in the - parameter space,
with additional emphasis in the above two asymptotic regimes.
In addition, for the asymptotic regime of low arrival, where for a
, the arrival rate
, stability is intuitive
fixed speed
as demands arrive very rarely. Hence, in this regime, we seek to
minimize the steady state expected delay for a demand.
B. The TMHP-Based Policy
We now present a novel receding horizon service policy for
the vehicle that is based on the repeated computation of a translational minimum Hamiltonian path through successive groups
of outstanding demands. For a given arrival rate and demand
speed
, let
denote the vehicle location in
the environment that minimizes the expected time to service
a demand once it appears on the generator. The expression
for
is postponed to Section IV-A. The TMHP-based
policy is summarized in Algorithm 1, and an iteration of the
policy is illustrated in Fig. 4.
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i) For a general
Algorithm 1: The TMHP-based policy
Assumes: The optimal location
1
if no outstanding demands are present in
is given.
then
ii) In the asymptotic regime of high arrival where
(and so
), the policy is stable if
2
3
else
4
5
A plot of the necessary and sufficient conditions is shown in
Fig. 1. In the asymptotic regime of high speed, the sufficient
condition from part (i) of Theorem III.2 simplifies to
Repeat.
C. Main Results
The following is a summary of our main results and the locations of their proofs within the paper. We begin with a necessary
condition on the problem parameters for stability of any policy,
the proof of which is presented in Section V-A. The condition
is policy independent and especially in the asymptotic regime
(and therefore
), we charof high speed, i.e.,
acterize the way in which
.
Theorem III.1 (Necessary Condition for Stability): The following are necessary conditions for the existence of a stabilizing
policy:
i) For a general
ii) For the asymptotic regime of high speed, where
,
every stabilizing policy must serve the demands in the
order in which they arrive. Further, the arrival rate must
tend to zero as per
Next, we present a sufficient condition on the problem parameters that ensures stability of the TMHP-based policy, the proof
of which is presented in Section V-B. We present a condition
for a general value of the demand speed and especially in the
(and thereasymptotic regime of high arrival, i.e.,
fore
), we characterize the way in which
and
, in order to ensure stability of the TMHP-based policy.
We first introduce the following notation. Let
for
,
otherwise
where
, and
is the solution
to
, and
is approximately equal to 2/3.
Theorem III.2 (Sufficient Condition for Stability): The following are sufficient conditions for stability of the TMHP-based
policy.
and the necessary condition established in part (ii) of Theorem
III.1 simplifies to
In the asymptotic regime of high arrival, the sufficient condition
,
from part (ii) of Theorem III.2 is
and the necessary condition established in part (i) of Theorem
.
III.1 is
Theorems III.1 and III.2 lead to the following corollary.
Corollary III.3 (Constant Factor Sufficient Condition): In the
asymptotic regime of
i) high speed, which is the limit as
, the ratio
.
ii) high arrival, which is the limit as
, the ratio
.
The third and final result shows optimality of the TMHPbased policy with respect to minimizing the expected delay in
the asymptotic regimes of low arrival and high speed.
Theorem III.4 (Optimality of TMHP-Based Policy): In the
asymptotic regimes of
i) low arrival, where
for a fixed
; and
(and therefore
);
ii) high speed, where
the TMHP-based policy serves the demands in the order in
which they arrive, and also minimizes the expected time to
service a demand.
In other words, the TMHP-based policy becomes equivalent
to a first-come-first-served (FCFS) policy in the above two
asymptotic regimes. Theorem III.4 is proven for the FCFS
policy in Section V-B. By the equivalence between FCFS and
the TMHP-based policies in the above two asymptotic regimes,
the result also holds for the TMHP-based policy.
To study the stability of the TMHP-based policy, we introduce
and analyze the FCFS policy in Section IV.
IV. THE FIRST-COME-FIRST-SERVED (FCFS) POLICY
In this section, we present the FCFS policy and establish some
of its properties, and use these properties as tools to analyze the
TMHP-based policy. In the FCFS policy, the service vehicle
uses constant bearing control and services the demands in the
BOPARDIKAR et al.: DYNAMIC VEHICLE ROUTING FOR TRANSLATING DEMANDS
Fig. 5. The FCFS policy. The vehicle services the demands in the order of their
arrival in the environment, using the constant bearing control.
order in which they arrive. If the environment contains no demands, the vehicle moves to the location
which minimizes the expected time to catch the next demand to arrive. This
policy is summarized in Algorithm 2.
Algorithm 2: The FCFS policy
Assumes: The optimal location
1
if no outstanding demands are present in
is given.
then
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We seek to place the vehicle at location that minimizes the
expected time to service a demand once it appears on the generator. Demands appear at uniformly random positions on the
generator and the vehicle uses the constant bearing control to
to reach
reach the demand. Thus, the expected time
a demand generated at position
from vehicle position
is given by
(3)
The following lemma characterizes the way in which this expectation varies with the position .
Lemma IV.2 (Properties of the Expected Time): The expected
is convex in , for all
.
time
Additionally, there exists a unique point
that minimizes
.
Proof: Regarding the first statement, it suffices to show that
is convex for all . To do this,
the integrand in (3),
with respect to and
consider the Hessian of
given by
2
3
else
4
5
Repeat.
Fig. 5 illustrates an instance of the FCFS policy. The following lemma establishes the relationship between the FCFS
policy and the TMHP-based policy.
Lemma IV.1 (Relationship Between TMHP and FCFS):
Given an arrival rate and a demand speed , if the FCFS
policy is stable, then the TMHP-based policy is stable.
Proof: Consider an initial vehicle position and a set of outstanding demands, all of which have lower -coordinates than
the vehicle. Let us compare the amount of time required to service the outstanding demands using the TMHP-based policy
with the amount of time required for the FCFS policy. Both policies generate paths through all outstanding demands, starting at
the initial vehicle location, and terminating at the outstanding
demand with the lowest -coordinate. By definition, the TMHPbased policy generates the shortest such path. Thus, the TMHPbased policy will require no more time to service all outstanding
demands than the FCFS policy. Since this holds at every iteration of the policy, the region of stability of TMHP-based policy
contains the region of stability for the FCFS policy.
In Sections IV-A–C, we analyze the FCFS policy. We then
combine these results with the above lemma to establish analogous results for the TMHP-based policy.
The first question is, how do we compute the optimal position
? This is answered in Section IV-A.
A. Optimal Vehicle Placement
In this subsection, we study the FCFS policy when
is fixed and
. In this regime, stability is not an issue as
demands arrive very rarely, and the problem becomes one of optimally placing the service vehicle (i.e., determining
in the statement of the FCFS policy).
. For
, the Hessian is positive semi-definite because its
determinant is zero and its trace is non-negative. This implies
is convex in for each
, from which the
that
first statement follows.
Regarding the second statement, since demands are uniformly randomly generated on the interval
, the optimal
horizontal position is
. Thus, it suffices to show
that
is strictly convex. From the
term of the Hessian we see that
is strictly
convex for all
. But, letting
and
we can write
The integrand is strictly convex for all
,
implying that
is strictly convex on the line
, and that the point
is the unique minimizer.
Lemma IV.2 tells us that there exists a unique point
which minimizes the expected travel time. In addition,
we know that
. Obtaining a closed form expression
for
does not appear to be possible. Computing the integral
in (3), with
, one can obtain
where
. For each value of and , this convex expression can be easily numerically minimized over , to obtain
. A plot of
as a function of for
is shown in
Fig. 6.
For the optimal position , the expected delay between a
demand’s arrival and its service completion is
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 11, NOVEMBER 2010
note that the only policy that ensures that a demand’s -coordinate never exceeds that of the service vehicle (i.e., that all demands remain below the service vehicle at all time) is the FCFS
policy. In what follows, we prove the proposition statement by
computing the expected time to travel between demands using
the FCFS policy. First, consider a vehicle location
and a demand location with initial location
, the minimum time in which the vehicle can reach the demand is given
by
(4)
Fig. 6. Optimal position Y of the vehicle that minimizes the expected distance
.
to a demand, as a function of v . The generator length W
=1
Thus, a lower bound on the steady-state expected delay of any
. We now characterize the steady-state expected
policy is
, as tends to zero.
delay of the FCFS policy
Lemma IV.3 (FCFS Optimality): Fix any
. Then in
the limit as
, the FCFS policy minimizes the expected
time to service a demand, i.e.,
.
Proof: We have shown how to compute the position
which minimizes (3). Thus, if the vehicle is located at
, then the expected time to service the demand is minimized.
, the probability that demand
arrives before
But, as
the vehicle completes service of demand and returns to
tends to zero. Thus, the FCFS policy is optimal as
.
Remark IV.4 (Minimizing the Worst-Case Time): Another
metric that can be used to determine the optimal placement
is the worst-case time to service a demand. Using an
argument identical to that in the proof of Lemma IV.3, it is possible to show that for fixed
, and as
, the
FCFS policy, with
, minimizes the
worst-case time to service a demand.
. Second, assume there are many
and is undefined if
outstanding demands below the service vehicle, and none
above. Suppose the service vehicle completed the service of
. Let us comdemand at time and position
pute the expected time to reach demand
, with location
. Since arrivals are Poisson, it follows that
, with probability one. To simplify notation,
and
.
we define
Then, from (4)
Taking expectation and noting that
and
Now, we note that
, that
constant independent of , and that
Thus
are independent
is a positive
, and for every
B. A Necessary Condition for FCFS Stability
In the previous subsection, we studied the case of fixed
and
. In this subsection, we analyze the problem when
, and determine necessary conditions on the magnitude of
that ensure the FCFS policy remains stable. To establish these
conditions we utilize a standard result in queueing theory (cf.
[25]) which states that a necessary condition for the existence
of a stabilizing policy is that
, where
is the
expected time to service a demand (i.e., the travel time between
demands). We begin with the following result.
Proposition IV.5 (Special Case of Equal Speeds): For
,
there does not exist a stabilizing policy.
Proof: When
, each demand and the service vehicle
move at the same speed. If a demand has a higher vertical position than the service vehicle, then clearly the service vehicle
cannot reach it. The same impossibility result holds if the demand has the same vertical position and a distinct horizontal
position as the service vehicle. In summary, a demand can be
reached only if the service vehicle is above the demand. Next,
This means that the necessary condition for stability, i.e.,
, is violated. Thus, there does not exist a
stabilizing policy.
Next we look at the FCFS policy and give a necessary condition for its stability.
Lemma IV.6 (Necessary Stability Condition for FCFS): A
necessary condition for the stability of the FCFS policy is
for
otherwise
,
, where is the Euler constant;
is the solution to the equation
, and is approximately equal to
.
Proof: Suppose the service vehicle completed the service
of demand at time at position
, and demand
is located at
. Define
where
and
BOPARDIKAR et al.: DYNAMIC VEHICLE ROUTING FOR TRANSLATING DEMANDS
and
time between demands is given by
. For
, the travel
(5)
Observe that the function
inequality leads to
is convex in
and
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where
is the 1st order Struve function and
is 1st order Bessel function of the 2nd kind [33]. Using
about
a Taylor series expansion of the function
, followed by a subsequent analysis of the higher order
terms, one can show that
. Jensen’s
where
, and is the Euler constant. This
inequality implies that (8) can be written as
Substituting the expressions for the expected values, we obtain
where we have used the fact that
From the necessary condition for stability, we must have
To obtain a stability condition on we wish to remove
the term. To do this, note that from (6) we have
and thus
from
,
By simplifying, we obtain
(6)
This provides a good necessary condition for low . Next, we
obtain a much better necessary condition for large . Since is
convex in
, we apply Jensen’s inequality to (5) to obtain
The necessary stability condition is
necessary condition for stability is
, from which a
(7)
. Now, the random variable
is diswhere
tributed exponentially with parameter
and probability density function
Solving for
when
, we obtain that
(9)
Un-conditioning (7) on
, we obtain
(8)
The right hand side can be evaluated using the software Maple
and equals
The condition
, implies that the above
. We now have two
bound holds for all
, and (9) which holds
bounds; (6) which holds for all
for
. The final step is to determine the values of for
which each bound is active. To do this, we set the right-hand side
of (6) equal to the right-hand side of (9) and denote the solution
. Thus, the necessary stability condition is given by (6)
by
when
, and by (9) when
.
C. A Sufficient Condition for FCFS Stability
In Section IV-B, we determined a necessary condition for stability of the FCFS policy. In this subsection, we will derive the
following sufficient condition on the arrival rate that ensures stability for the FCFS policy. To establish this condition, we utilize
a standard result in queueing theory (cf. [25]) which states that
a sufficient condition for the existence of a stabilizing policy is
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that
, where
is the expected time to service a
demand (i.e., the travel time between demands).
Lemma IV.7 (Sufficient Stability Condition for FCFS): The
FCFS policy is stable if
for
where
, and
denotes
the Bessel function of the first kind. To obtain a lower bound on
, we observe that if
, then due to the term
, we can substitute
in
in the above lower bound for
place of
. Thus, we obtain
,
otherwise
(13)
where
, and
is the solution
to
, and
is approximately equal to 2/3.
Proof: We begin with the expression for the travel time
between two consecutive demands using the constant bearing
control (cf. Definition II.1)
(10)
where we used the inequality
. Thus
since the demands are distributed uniformly in the -direction
and Poisson in the -direction. A sufficient condition for stability is
Substituting into (12), we obtain
which yields
(14)
Now, let be the least upper bound on for which the FCFS
, the FCFS policy is
policy is unstable, i.e., for every
stable. To obtain , we need to solve
. Using (14),
we can obtain a lower bound on
by simplifying
(11)
The upper bound on given by (10) is very conservative except
for the case when is very small. Alternatively, taking expected
value of conditioned on
, and applying Jensen’s inequality
to the square-root part, we obtain
From the condition given by (11), the second term in the
parentheses satisfies
Thus, we obtain
since
. Following steps which are similar to
those between (7) and (8), we obtain
(12)
In [33], polynomial approximations have been provided
for the Struve and Bessel functions in the intervals
and
. We seek an upper bound for the right-hand side of
(12) when is sufficiently large, i.e., when the argument of
and
is small. From [33], we know that
where
. Since
stability, a sufficient condition for stability is
implies
(15)
To determine the value of the speed
beyond which this is
a less conservative condition than (11), we solve
For
, one can verify that the numerical value of the
argument of the Struve and Bessel functions is less than 2, and
BOPARDIKAR et al.: DYNAMIC VEHICLE ROUTING FOR TRANSLATING DEMANDS
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that at time ,
contains points in
(that is, the probability that
). We have
Since the generation process is temporally Poisson and spatially
uniform, the above equation can be rewritten as
(16)
Fig. 7. Necessary and sufficient conditions for the stability for the FCFS
policy. The dashed curve is the necessary condition for stability as established
in Lemma IV.6; while the solid curve is the sufficient condition for stability as
established in Lemma IV.7.
so the bounds given by (13) used in this analysis are valid. Thus,
a sufficient condition for stability is given by equation(11) for
, and by (15) for
.
Remark IV.8 (Tightness in Limiting Regimes): As
,
the sufficient condition for FCFS stability becomes
,
which is exactly equal to the necessary condition given in
Lemma IV.6. Thus, the condition for stability is asymptotically
tight in this limiting regime.
Fig. 7 shows a comparison of the necessary and sufficient stability conditions for the FCFS policy. It should be noted that
converges to
extremely slowly as tends to
, and still
satisfy the sufficient stability condition in Lemma IV.7. For ex, the FCFS policy can stabilize the
ample, with
.
system for an arrival rate of
V. PROOFS OF THE MAIN RESULTS
Now we compute
and
so that, substituting these expressions and adopting the shortand
, (16) becomes
hands
Rewriting
of binomial
as
(17)
, and using the definition
, (17) reads
In this section, we present the proofs of the main results which
were presented in Section III-C.
A. Proof of Theorem III.1
We first present the proof of part (i). We begin by looking at
the distribution of demands in the service region.
Lemma V.1 (Distribution of Outstanding Demands): Suppose the generation of demands commences at time 0 and no
demands are serviced in the interval
. Let denote the set
at time . Then, given a Borel
of all demands in
measurable set of area contained in
Proof: We first establish the result for a rectangle. Let
be a rectangle contained in
with area
. Let us calculate the probability
Finally, since
, we obtain
where
. Thus, the result is established for a closed
rectangle. But notice that the result also holds for the case when
a rectangle is open. Further, the result holds for a countable
union of disjoint rectangles due to the demand generation
process being uniform along the generator and Poisson in time.
can be generated
Now, a Borel measurable set on
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by countable unions and complements of rectangles. But both,
the unions and the complements of rectangles can be written as
a disjoint union of a countable number of rectangles. Hence, the
.
result holds for any Borel measurable set in
Remark V.2 (Uniformly Distributed Demands): Lemma V.1
shows that the number of demands in an unserviced region
,
with area is Poisson distributed with parameter
and conditioned on this number, the demands are distributed
uniformly.
Lemma V.3 (Travel Time Bound): Consider the set of demands that are uniformly distributed in at time . Let
be a
random variable giving the minimum amount of time required to
.
travel to a demand in from a vehicle position selected
Then
Proof: Let
denote the vehicle location selected a priori. To obtain a lower bound on the minimum travel
time, we consider the best-case scenario, when no demands have
. Consider a demand in
been serviced in the time interval
with position
at time . Using Proposition II.2, we can
write the travel time from to
implicitly as
(18)
as the collection of demands that can be
Next, define the set
reached from
in or fewer time units. From (18) we
see that when
, the set
is a disk of radius centered at
. That is
where we have omitted the dependence of on and . The
area of the set
, denoted
, is upper bounded by
, and the area is equal to
if the
does not intersect
a boundary of . Now, by Lemma V.1 the demands in an unserviced region are uniformly randomly distributed with den. Let us compute the distribution of
sity
. For every vehicle position chosen before
the generation of demands, the probability that
is given
by
where the second equality is by Lemma V.1, and the last inequality comes from the fact that
(Note that
this is equivalent to assuming that the entire plane
contains
demands with density ). Hence, we have
We can now prove part (i) of Theorem III.1.
Proof of part (i) of Theorem III.1: A necessary condition
for the stability of any policy is
where
demands
is the steady-state expected travel time between
and
. For every policy
. Thus, a necessary condition for stability is
that
Remark V.4 (Constant Fraction Service): A necessary condition for the existence of a policy which services a fraction
of the demands is that
Thus, for a fixed
no policy can service a constant
. This follows because in
fraction of the demands as
.
order to service a fraction we require that
In order to service a fraction of the demands, we consider a
, with the arrival rate
subset of the generator having length
on that subset being equal to . The use of the TMHP-based
policy on this subset and with the arrival rate gives a sufficient
condition for stability analogous to Theorem III.2, but with an
extra term of in the denominator.
For the proof of part (ii) of Theorem III.1, we first recall from
Lemma IV.6 that for stability of the FCFS policy, although
must go to zero as
, it can go very slowly to 0. Specifically, goes to zero as
This quantity goes to zero more slowly than any polynomial in
. We are now ready to prove Theorem III.1.
Proof of part (ii) of Theorem III.1: The central idea of this
, if the vehicle skips a deproof is that in the limit as
mand and services the next one at an instance, then to reach the
demand it skipped, the vehicle has to travel infinitely far from
the generator which leads to instability.
Observe that the condition on in the statement of part (ii)
is the expression given by the necessary condition for FCFS
, from Lemma
stability in the asymptotic regime as
IV.6. Therefore, suppose there is a policy that does not serve
demands FCFS, but can stabilize the system with
for some
, and
. Let be the first instant at which
policy deviates from FCFS. Then, the demand served immediately after is demand
for some
. When the vehicle
reaches demand
at time
, demand
has moved above
the vehicle. To ensure stability, demand
must eventually
be served. The time to travel to demand
from any demand
, where
, is
BOPARDIKAR et al.: DYNAMIC VEHICLE ROUTING FOR TRANSLATING DEMANDS
where
and
are now the minimum of the and distances
from
to the
. The random variable
is Erlang
distributed with shape
and rate , implying
for each
, and in particular, for
. Now, since
as
,
, with probability one. Thus
with probability one, as
demands that arrive during
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where
is the
-coordinate of
and
is the time taken for
, serves all
the vehicle as per the TMHP that begins at
and ends at
.
demands in other than
of the TMHP for
We seek an upper bound for the length
which we use the Convert-to-EMHP method (cf. Section II-B).
, and writing
Invoking Lemma II.5 for
for convenience, we have
. Thus, the expected number of
is
as
. This implies that with probability one, the policy
becomes unstable when it deviates from FCFS and that any
. Thus,
deviation must occur with probability zero as
a necessary condition for a policy to be stabilizing with
is that, as
, the policy must serve demands in
the order in which they arrive. But this needs to hold for every
and, by letting go to infinity,
converges to zero for all
. Thus, a non-FCFS policy cannot stabilize the system
as
. Hence, as
,
no matter how quickly
every stabilizing policy must serve the demands in the order
in which they arrive. Additionally, notice that the definition of
the FCFS policy is that it uses the minimum time control (i.e.,
constant bearing control) to move between demands, thus the
expression in part (ii) of Theorem III.1 is a necessary condition
.
for all stabilizing policies as
B. Proofs of Theorem III.2 and Theorem III.4
We first present the proof of Theorem III.2.
Proof of part (i) of Theorem III.2: The central idea is to derive a recurrence relation between the expected coordinate of
the vehicle at the next iteration and the expected coordinate at
the present iteration by using the background results on the computation of the TMHP in the iteration. The stability condition
follows by ensuring that the resulting evolution of the expected
coordinate with the number of iterations remains bounded.
Note that if there are any demands “above” the vehicle initially, at the end of the first iteration of the TMHP-based policy,
all outstanding demands have their y-coordinates less than or
equal to that of the vehicle, and hence would be located “below”
the vehicle as shown in the first of Fig. 4. Hence at the end of
every iteration of the TMHP-based policy, all outstanding demands would be located “below” the vehicle.
and
Let the vehicle be located at
denote the demand with the least y-coordinate at time instant .
Let
denote the number of demands in the set . If there
exists a non-empty set of unserviced demands below the ve, with probability
hicle at time , then for each
where the first inequality is due to Lemma II.3, and the second
. Thus, when is non-empty at time
is due to
If is empty at time , then the vehicle moves towards the
. When a new demand arrives, the
optimal location
, then in the worst-case,
vehicle moves towards it. If
the vehicle is very close to an endpoint of the generator and
the next demand arrives at the other endpoint. In this case, the
vehicle moves with a vertical velocity component equal to and
horizontal component equal to
. So in the worst-case,
at the beginning of the
the vehicle is at a height
. In this case,
next iteration. The other possibility is if
to get an upper bound on the height of the vehicle at the next
iteration, we consider the vehicle motion when it first moves
horizontally so that the -coordinate equals that of the demand,
and then moves vertically down to meet the demand. This gives
an upper bound on the height of the vehicle at the next iteration
. Thus, if is empty, then the sum of
as
these two upper bounds is trivially an upper bound on the height
of the vehicle at the beginning of the next iteration. Thus, if
is empty, then
Conditioned on
, we have
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It can be shown that
. Collecting the terms
together and on further simplifying,
with
. Moreover, we may upper bound the
bound for
expected delay for a demand by
where the equality follows since the arrival process is Poisson
with rate and for a time interval
. Substituting into (19),
we obtain
Proof of part (ii) of Theorem III.2: As the arrival rate
, the necessary condition in Theorem III.1 states that for
stability, must tend to zero. Now, for this part, we make use
, the length of the
of the following two facts. First, as
TMHP constrained to start at the vehicle location and end at
the lowest demand, is equal to the length of the EMHP in the
corresponding static instance under the map , as described in
of points which are
Lemma II.5. Second, consider a set
uniformly distributed in a region with finite area. Then, in the
, the length of a constrained EMHP through
limit as
tends to the length of the ETSP tour through .
More specifically, consider the th iteration of the TMHPbased policy, and let
be the position of the service
, the number of outstanding
vehicle. In the limit as
, and in addition, condidemands in that iteration
tioned on , the demands are uniformly distributed in the re(cf. Remark V.2). Now using the above
gion
two facts, we can apply Theorem II.4 to obtain an expression
for the length of the TMHP constrained to start at the vehicle
, the
location and ending at the lowest demand. As
position of the vehicle at the end of the th iteration satisfies
Using the law of iterated expectation, we have
with probability one, where
is the area of the region
below the vehicle at the th iteration. Thus, conditioned on
being bounded away from 0, we have
(19)
Applying Jensen’s inequality to the conditional expectation in
the second term in the right hand side of (19)
with probability one, where we have applied Jensen’s in. Thus,
equality. Using Lemma V.1,
with probability one
(20)
Thus,
is finite if
Thus, if satisfies the above condition, then expected number
of demands in the environment is finite and the TMHP-based
policy is stable.
Finally, from Lemma IV.1, the region of stability for the
FCFS policy is contained in the region of stability for the
TMHP-based policy. Thus, the TMHP-based policy is stable
for all arrival rates for which the FCFS policy is stable. Thus,
the TMHP-based policy is stable for all arrival rates satisfying
the bound in Lemma IV.7. This gives us the desired result.
Remark V.5 (Upper Bound on Expected Delay): Since (20)
, we can easily obtain an upper
is a linear recurrence in
Thus, the sufficient condition for stability of the TMHP-based
(and thus
) is
policy as
Finally, we present the proof of Theorem III.4.
Proof of Theorem III.4: It follows from Lemma IV.3 and
Lemma IV.1 that the FCFS minimizes the steady state expected
delay in the asymptotic regime of low arrival. Part (i) of Theorem III.4 follows since in the asymptotic regime of low arrival,
the TMHP-based policy becomes equivalent to the FCFS policy.
The proof of part (ii) follows from part (ii) of Theorem III.1 and
Lemma IV.1 along with the fact that the TMHP-based policy
spends the minimum amount of time to travel between demands.
BOPARDIKAR et al.: DYNAMIC VEHICLE ROUTING FOR TRANSLATING DEMANDS
Fig. 8. Numerically determined region of stability for the TMHP-based policy.
A lightly shaded (green-colored) dot represents stability while a darkly shaded
(blue-colored) dot represents instability. The uppermost (thick solid) curve is
the necessary condition for stability for any policy as derived in Theorem III.1.
The lowest (dashed) curve is the sufficient condition for stability of the TMHPbased policy as established by Theorem III.2. The broken curve between the two
curves is the sufficient stability condition of the TMHP-based policy in the high
arrival regime as derived in part (ii) of Theorem III.2. The environment width
.
is
W =1
VI. SIMULATIONS
In this section, we numerically determine the region of stability of the TMHP-based policy, and compare it with the theoretical results from the previous sections.
In the actual implementation of the TMHP-based policy, the
computational complexity increases undesirably as the values of
approach the instability region.
the problem parameters
Therefore, we adopt a different procedure to characterize the
stable/unstable region boundary, based upon the following central idea. For a given value of
and a sufficiently high value
of the height of the vehicle, if the policy is stable, then after one
iteration of the policy, the vehicle’s height must decrease. In particular, the following procedure was adopted.
i) For a collection of instructive pairs of the demand speed
and in the region of interest, we set the generator width
and we set the initial height of the environment
of interest so that the expected number of demands in the
environment are 1000. Thus,
.
ii) We repeated 10 times the following procedure. The vehicle is placed at the height and at a uniformly random
location in the horizontal direction. A Poisson distributed
number
with parameter
, of outstanding demands
are uniformly randomly placed in the environment (cf.
Lemma V.1). The vehicle uses the TMHP-based policy
to serve all outstanding demands and we store the height
of the vehicle at the end of the single iteration of the
policy. Finally, we compute the average height
of the
10 iterations.
iii) If
, then the policy is deemed to be stable for the
chosen value of
. Otherwise the policy is deemed to
be unstable.
1 solver was used to generate approximations
The
to the TMHP at every iteration of the policy. The linkern solver
takes as an input an instance of the ETSP. To transform the constrained EMHP problem into an ETSP, we replace the distance
1The TSP solver linkern is freely available for academic research use at
http://www.tsp.gatech.edu/concorde.html.
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v Fig. 9. Variation of the steady state vehicle height with the product 1 at
different demand speeds. The error bars correspond to 6 standard deviation
about the mean.
1
between the start and end points with a large negative number,
ensuring that this edge is included in the linkern output.
The results of this numerical experiment are presented in
Fig. 8. For the purpose of comparison, we overlay the plots
for the theoretical curves, from Fig. 1. We observe that the
numerically obtained stability boundary for the TMHP-based
policy falls between the necessary and the sufficient conditions
which were established in parts (i) of Theorems III.1 and
III.2 respectively. Further, although the sufficient condition
characterized in part (ii) of Theorem III.2, is theoretically
an approximation of the stability boundary in the asymptotic
regime of high arrival, our numerical results show that the
condition serves as a very good approximation to the stability
boundary, for nearly the entire range of demand speeds.
in the stable region, we present
For different values of
simulations of the TMHP-based policy that shed light on the
steady state height of the vehicle. For three different values of
, we empirically determine the steady state vehicle height at
various arrival rates. The results are shown in Fig. 9. We observe
that the steady state height increases i) at higher arrival rates for
a fixed demand speed, and ii) at higher demand speeds at fixed
arrival rates, which is fairly intuitive to expect for a stable policy.
VII. CONCLUSION
We introduced a dynamic vehicle routing problem with translating demands. We determined a necessary condition on the arrival rate of the demands for the existence of a stabilizing policy.
In the limit when the demands move as fast as the vehicle,
we showed that every stabilizing policy must service the demands in the FCFS order. We proposed a novel receding horizon
policy that services the moving demands as per a translational
minimum Hamiltonian path. In the asymptotic regime when
the demands move as fast as the vehicle, we showed that the
TMHP-based policy minimizes the expected time to service a
demand. We derived a sufficient condition for stability of the
TMHP-based policy, and showed that in the asymptotic regime
of low demand speed, the sufficient condition is within a constant factor of the necessary condition for stability. In a third
asymptotic regime when arrival rate tends to zero for a fixed demand speed, we showed that the TMHP-based policy is optimal
in terms of minimizing the expected time to service a demand.
Finally, we presented an implementation of the TMHP-based
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where the notation
denotes the shortest distance
of point
from the nearest of the
lines. The
second path is constructed similarly using the
lines
. This path also
commences on
, passes through the above lines (visiting the points whenever they are at the shortest distance from
these lines) and ends on
. The length of this path is
where the notation
denotes the shortest distance of point
from the nearest of the new lines.
Observe that
. Hence
Fig. 10. Illustration of the proof of Lemma II.3. 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, visiting a point whenever it is within a distance of h=2m for a specific
integer m from the solid horizontal lines.
2
policy to numerically determine its region of stability. Our numerical results show that the theoretically established sufficient
condition in the asymptotic regime of low demand speeds also
serves as a good approximation to the boundary of the stability
region for a significantly large interval of values of demand
speed.
As future work, it would be of interest to extend the present
analysis or to design an improved policy that is optimal also
in the asymptotic regime of high arrival, in addition to the two
other asymptotic regimes. Another interesting direction is to
have on-site service times for the demands. For the case in which
the on-site service times are independent and identically distributed with a known expected value, and the vehicle is permitted to move with the demand upon reaching it, the results in
this paper could be extended using similar analysis. Recently,
in [34], we addressed a version of the present problem in which
the goal for the vehicle is to maximize the fraction of demands
served before they reach a deadline, which is at a given distance
from the generator. In [35], we addressed a vehicle placement
problem that involves non-uniform spatial arrival density and
also multiple service vehicles.
APPENDIX
In this Appendix, we present the proof of Lemma II.3.
Proof: Suppose the rectangular region is given by
,
. Let be a positive integer (to be chosen later) and
let the points be denoted by
. We now construct
two paths through the points. The first consists of (a) the
lines
; (b) the shortest distances
from each of the points to the nearest such line, each traveled
twice, and (c) suitable portions of the lines
,
,
and
,
. This is illustrated in Fig. 10. The length
of this path is
Now choose
to be the integer nearest to
, where
. Thus
, so that
Thus, at least one of the two paths must have length upper
.
bounded by
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Shaunak D. Bopardikar (S’08–M’10) received
the B.Tech. and M.Tech. degrees in mechanical
engineering from Indian Institute of Technology,
Bombay, in 2004, and the Ph.D. degree in mechanical engineering from the University of California at
Santa Barbara, in 2010.
From 2004 to 2005, he was an Engineer with
General Electric India Technology Center, Bangalore, India. He is currently a Postdoctoral Researcher
with the Center for Control Dynamical Systems
and Computation (CCDC), University of California
at Santa Barbara. His research interests include large-dimensional dynamic
optimization, game theory, pursuit-evasion, and motion coordination for robotic
systems.
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Stephen L. Smith (S’05–M’09) received the B.Sc.
degree in engineering physics from Queen’s University, Kingston, ON, Canada, in 2003, the M.A.Sc. degree in electrical engineering from the University of
Toronto, Toronto, ON, Canada in 2005, and the Ph.D.
degree in mechanical engineering from the University of California, Santa Barbara, in 2009.
He is currently a Postdoctoral Associate in the
Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology (MIT),
Cambridge. His research focuses on the distributed
control of autonomous systems. Particular interests include persistent monitoring, vehicle routing, and task allocation in dynamic environments.
Dr. Smith was a finalist for the Student Best Paper Award at the IEEE Conference on Decision and Control in 2007.
Francesco Bullo (S’95–M’99–SM’03–F’10) received the Laurea degree (with highest honors) in
electrical engineering from the University of Padova,
Padova, Italy, in 1994, and the Ph.D. degree in
control and dynamical systems from the California
Institute of Technology, Pasadena, in 1999.
From 1998 to 2004, he was an Assistant Professor
with the Coordinated Science Laboratory, University
of Illinois at Urbana-Champaign. He is currently a
Professor with the Mechanical Engineering Department and the Center for Control, Dynamical Systems
and Computation, University of California, Santa Barbara. He is the coauthor
of Geometric Control of Mechanical Systems (Springer, 2004) and Distributed
Control of Robotic Networks (Princeton Univ. Press, 2009). He has published
more than 150 papers in international journals, books, and refereed conferences.
He has served, or is serving, on the editorial boards of the ESAIM: Control, Optimization, and the Calculus of Variations and the SIAM Journal of Control and
Optimization. His main research interest is multi-agent networks with application to robotic coordination, distributed computing and power networks; he has
worked on problems in vehicle routing, geometric control, and motion planning.
Dr. Bullo received the 2003 ONR Young Investigator Award and the 2008
Outstanding Paper Award for the IEEE Control Systems Magazine. He served
on the editorial board of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL.
João P. Hespanha (M’94–F’08) was born in
Coimbra, Portugal, in 1968. He received the Licenciatura in electrical and computer engineering from
the Instituto Superior Técnico, Lisbon, Portugal in
1991 and the Ph.D. degree in electrical engineering
and applied science from Yale University, New
Haven, CT, in 1998.
From 1999 to 2001, he was Assistant Professor at
the University of Southern California, Los Angeles.
He moved to the University of California, Santa Barbara in 2002, where he currently holds a Professor
position with the Department of Electrical and Computer Engineering. He is
Associate Director for the Center for Control, Dynamical-systems, and Computation (CCDC), Vice-Chair of the Department of Electrical and Computer Engineering, and a member of the Executive Committee for the Institute for Collaborative Biotechnologies (ICB). His current research interests include hybrid and
switched systems; the modeling and control of communication networks; distributed control over communication networks (also known as networked control systems); the use of vision in feedback control; and stochastic modeling in
biology.
Dr. Hespanha received the Yale University’s Henry Prentiss Becton Graduate Prize for exceptional achievement in research in Engineering and Applied
Science, a National Science Foundation CAREER Award, the 2005 best paper
award at the 2nd International Conference on Intelligent Sensing and Information Processing, the 2005 Automatica Theory/Methodology best paper prize, the
2006 George S. Axelby Outstanding Paper Award, and the 2009 Ruberti Young
Researcher Prize. From 2004 to 2007, he was an Associate Editor for the IEEE
TRANSACTIONS ON AUTOMATIC CONTROL. He is an IEEE Distinguished Lecturer since 2007.