IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4, NO. 5 , OCTOBER 1988 49 1

Traffic Control of Multiple Robot Vehicles

Abstract-This paper addresses the control of traffic when there are large numbers of Automatic Guided Vehicles (AGV’s) in a factory. Although in most existing systems, AGV’s follow predetermined tracks with few switches, newer AGV technology will allow each vehicle to follow unconstrained paths. This paper analyzes several possible policies for managing AGV’s to minimize the impact of traffic jams. In a grid- iron network of roads, it is shown that traffic contention can have a large effect on the collective performance of AGV’s. It is shown that a particular traffic policy permits the AGV’s to be autonomous and independent while avoiding deadlocks. This policy is shown to be close to optimal, even for large numbers of vehicles. It is also shown under what conditions it is beneficial to modify the factory to have highways and superhighways.

INTRODUCTION UTOMATIC GUIDED VEHICLES (AGV’s) are being A used increasingly as a means of delivering parts and

materials within highly automated factories [3], [ l l ] . In most existing AGV installations, the traffic flow is simple. The vehicles follow predetermined routes, specified by buried cables or chemical lines on the floor. The roads are all one- way, vehicles are not permitted to overtake one another, and there are few forks and joins, except for spurs to service workstations. Even with this restricted topology, the determi- nation of optimal layouts, routes, and schedules is nontrivial

As AGV technology progresses, it will become possible to allow vehicles to travel much more freely in two dimensions. This possibility has stimulated research on many topics dealing with individual AGV’s: planning paths to avoid stationary obstacles [9], navigation [4], and control, including issues of sensing, kinematics, and dynamics.

As research progress is made on reliable navigation and control of individual vehicles, it will become possible to design future factories to contain large numbers of AGV’s, each with the freedom to navigate in two dimensions over a network of roads having a large number of forks and joins. This possibility raises a new technical problem: AGV traffic control.

The problem of AGV traffic control has not been addressed in either the technical or trade press. This problem is superficially similar to that of controlling automobile traffic in a city, a topic for which there is an extensive technical literature [6], [7], [ 141 covering both theory and application. There is, however, an important difference between autos and AGV’s: For traffic in a city, one expects each driver to

[lo].

Manuscript received November 20, 1986; revised January 7, 1988. The author is with Stanford Center for Integrated Systems, Palo Alto, CA

94303, on leave from the Manufacturing Research Department, IBM T. J. Watson Research Center, Yorktown Heights, NY.

IEEE Log Number 8821747.

selfishly optimize his individual travel time. In the factory domain, on the other hand, one expects all AGV’s to cooperate to serve the common good.

The freedom to have AGV’s choose optimal routes in a road network offers three benefits over predetermined paths with few forks and joins. First, optimal routes are usually shorter than serpentine one-way paths. Thus for a given factory throughput and given number of AGV’s, the speed of the AGV’s can be decreased. Second, it is easier to modify a factory layout without causing major perturbations. For example, one section of roadway can be blocked off to permit AGV repair or roadway maintenance, without deadlocking the traffic. Third, it facilitates higher job mix and more rapid change of jobs in a job-shop environment, without the need to modify the layout of machine tools.

All of these advantages will be lost if AGV traffic is controlled poorly. Without good control procedures, traffic can be hopelessly inefficient, even to the point of total deadlock. The purpose of this paper, therefore, is to explore several possible policies for managing AGV’s to assure efficient traffic flow.

ALTERNATNE METHODS OF TRAFFIC CONTROL

There are three basic alternative approaches to controlling AGV traffic:

1) The roads are restricted so that there is a unique route from any point to any other point. This is the case for nearly all existing systems.

2) There are multiple possible routes, and each AGV has the autonomy to select its own route, in accordance with benevo- lent policies intended to serve the common good.

3) There are multiple possible routes, and all AGV’s are under centralized traffic control. The level of control may range from simple traffic lights to global routing optimization. The latter extreme involves very high combinatorial complex- ity.

Since any roadway restriction (Case 1) can be recast in the form of autonomous AGV route selection constraints (Case 2), it is clear that the optimal solution under Case 2 is always at least as good as that under Case 1. Similarly, since the centralized route planner (Case 3) can always be restricted to perform precisely the autonomous planning (Case 2), it is clear that the optimal solution under Case 3 is always at least as good as that under Case 2.

This paper introduces a toy problem in AGV traffic control, intended to be sufficiently simple that it permits analysis without being so simplistic that it is completely unrepresenta- tive of the real world of manufacturing. Case 1 is briefly analyzed as a preliminary to the main section of the paper,

08824967/88/1000-0491$01 .OO 0 1988 IEEE

492 IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4 , NO. 5 , OCTOBER 1988

Fig. 1. Grid-iron floor plan.

which is a detailed study of Case 2. Case 3 will not be considered further.

The formulation of the problem and the methods of analysis presented in this paper are different from those found in the previously existing technical literature.

TOY PROBLEM Consider a factory whose floor plan is a square grid-iron

with n parallel roads along each axis. In the graph shown in Fig. 1, the 2n ( n - 1) arcs represent

allowed AGV paths, and the n2 nodes represent locations of machine tools to be serviced. It is assumed that the service consists of moving parts among the nodes in accordance with predetermined uniformly distributed random patterns. Al- though in the real world a part would never be moved directly from any machine to the same machine, this possibility is permitted here because it simplifies the analysis in a later section.

The number of AGV’s on the floor is r , where 1 5 r I n2 - 1. All vehicles move at the same speed U , traversing one arc every l / u units of time. The service time for loading and unloading machine tools is assumed to be negligible in comparison to the transit time for moving among tools,

AGV’s are assumed to be continually transporting parts between random pairs of machines. Each time that an AGV completes a delivery, it is assigned a new source and sink pair. Because each AGV has only one task to perform, this problem differs from the typical routing problem in which the “traveling salesman” knows many tasks in advance [ 2 ] . To keep the analysis simple, it is further assumed that AGV’s have no memory of how or why they arrived at their current locations, and that AGV’s can never trade tasks with one another, even though a strategy of trading tasks could increase throughput.

Once an AGV is given a task, it travels to the source, picks up a part, travels to the sink, and delivers the part. In this process, a number of time steps elapse, during each of which the AGV traverses either no arcs or one arc of the graph. The average throughput W of all the AGV’s together is

u r W = - S

. . . . . . . . . . . . . . . . . . . . . A A A + +<<<<<<<+<<< ,

A V A V A V + +>>>>>>>+>>> . A A A

.. >>>+>>>>>>>+ V V v

.. <<<+<<<<<<<+

. . >>>+>>>>>>>+ V v V

A V A V A V + +>>>>>>>+>>> . , . >>>+>>>>>>>+ A V A V A v +<<<<<<<+<<<<<<<+<<< . _ . <<<+<<<<<<<+

Fig. 2. One-way floor plan.

where S is the average number of time steps per task for each AGV .

This throughput must exactly match the throughput of all the machine tools. It is assumed that small buffers at the machines are used to smooth the surges in activity. The throughput of the machine tools is proportional to the number of machines n 2 , and there is no loss of generality in assuming that the constant of proportionality is 1. Thus the AGV speed must satisfy

Sn r

U = - ,

It is also assumed that the price of any number of AGV’s is completely negligible in comparison to the price of each machine tool. Therefore, management is willing to put an arbitrarily large number r of AGV’s on the factory floor in order to minimize U. For a given factory of size n x n, therefore, the problem is one of optimizing the traffic control and the value r in order to minimize U.

The traffic rules are the following:

At the end of each step at most one AGV may be at each

During each step no two AGV’s may pass on the same

All AGV’s have equal priority.

node.

arc.

CASE 1 : RESTRICTED ROADWAYS

The restricted roadways approach is to block off n 2 - 2n arcs to leave n 2 arcs along a single serpentine one-way closed path through all n2 nodes, as shown in Fig. 2. Such a path can always be constructed if and only if n is even, in which case there are a large number of alternative paths like the one in the figure.

For this case, there is never any traffic problem. Each AGV makes a complete circuit every n 2 steps. At the moment an AGV is assigned a random task, there is an average latency of ( n 2 - 1)/2 steps until it will reach the source and then another ( n 2 - 1)/2 steps until it will reach the sink. Thus each AGV completes a task every n 2 - 1 steps.

GROSSMAN: TRAFFIC CONTROL OF ROBOT VEHICLES 493

Fig. 3. Never backup deadlock. Each vehicle Vli is trying to get to target Tli.

Since the r AGV's are all independent, the AGV speed is

n 2 ( n 2 - 1) r

given by

U =

This speed is minimized by letting r be as large as possible. Setting r equal to its largest possible value of n2 - 1 gives the optimal

u = n 2 . (2)

As the factory is made arbitrarily large, the optimal AGV speed increases as n2 . This fact is a blatant indication that the restricted roadway approach to AGV traffic control is abys- mally bad for the given set of assumptions. This is the motivation for progressing to Case 2.

CASE 2: AUTONOMOUS ROUTE SELECTION

To resolve traffic conflicts, the AGV's are assigned priorities from 1 to m, but these priorities are rotated in successive time steps to assure that on the average all vehicles have equal priority. At each time interval, in order of current priority, each AGV plans a feasible next step and then reserves that step. A step may be either moving to an adjacent node or remaining stationary at the current node.

In planning a feasible next step, each AGV attempts to get closer to its next target node. In general, if the target is in a different row and column, the AGV will have two alternative arcs that are equally good. If the number of vehicles is large, however, one or both of these arcs may be blocked.

When an AGV is blocked, various policies of waiting, side- stepping, or backing-up can be assessed in terms of their ability to optimize throughput by avoiding deadlock and delay. Even when an AGV is not blocked, policies for choosing between two alternative arcs in the right direction can be assessed in terms of their ability to avoid blocking lower priority AGV's. Any of these policies may be either determi- nistic or stochastic.

For example, consider the policy that any blocked AGV will always wait. Clearly, even for only 2 AGV's in a large empty factory, this policy can lead to deadlock. Somewhat less obvious is the fact that a never backup policy can also result in deadlock. An example is the situation of 8 AGV's on a 4 x 4 grid as shown in Fig. 3.

Two sets of policies are now considered in greater depth. The first of these is the obvious greedy policy of having each AGV try to follow a path which at each step heads in the

V I 3

Fig. 4. Greedy policy after 1, 2, and 3 steps.

direction that is most nearly towards its target. This policy yields a path that is a straight line on the grid followed by a 45' zig-zag. The second policy is less obvious but more benevo- lent; it tries to encourage an overall counterclockwise flow of traffic. It will be shown that the second policy is far superior.

Greedy Policy Suppose that each AGV follows the following rules:

1) Among all unblocked immediately adjacent nodes, make a step to the node that is closest to the target.

2) If 2 nodes are equally close, choose randomly between them.

3) If all adjacent nodes are blocked, then wait at the current node.

Rule 1 actually requires an AGV to back up if it is blocked from getting nearer its target but has immediately adjacent unblocked nodes that are further from its target. In the situation shown in Fig. 3, these rules would cause AGV's 1 and 2 to back up, after which 3 and 4 could move down. AGV 5 could then move left. AGV 6 would back up, leaving room for 7 to move up, and finally 8 would move left into the space vacated by 4. The resulting situation after one step is shown in Fig. 4. The priorities would then be rotated so that AGV 2 would plan its motion first, and so forth.

Subject to some random choices between equally good

494 IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4, NO. 5 , OCTOBER 1988

nodes, the subsequent states might be those shown in Fig. 4. It can be seen that only AGV 3 has reached its target, and that in general many of the AGV motions seem to be churning almost aimlessly.

One AGV: The performance of only one AGV can be derived analytically as follows:

First, consider motions on a one-dimensional grid. Suppose i a n d j are random rows on this grid. Then 1 5 i I n and 1 I j 5 n. The number of steps between i and j is I i - j I . The expectation value of I i - j I is therefore given by

where = a ( a - 1). . - ( a - k + 1). From the calculus of finite differences, the above expression is easily summed to give

n 2 - 1

3n

On a two-dimensional grid, since the distribution for target rows is independent of that for columns, the expected number of steps between any two random locations is

2(n2- 1) 3n

Since a task consists of moving to a source and then to a sink, the expected number SI of steps per task is

0.0052 0.0081 0.0093 0.0098 0.0098 0.0093 0.0081 0.0052

0.0081 0.0144 0.0175 0.0186 0.0186 0.0175 0.0144 0.0081

0.0093 0.0175 0.0231 0.0255 0.0255 0.0231 0.0175 0.0093

4 ( n 2 - 1) 3n

s*=-.

is true, it is reasonable to make the approximation that to first order the behavior of 1 factory with r AGV's can be approximated by superimposing r factories with 1 AGV each. The probability P k ( i , j ) that there are exactly k AGV's at node ( i , j ) of the superimposed factory is given approximately by a Poisson distribution [ 5 ] .

where h = rF.

the expected number of AGV's in contention at node (i, j ). Whenever k is 0 or 1, there is no contention. Let C ( i , j ) be

- Xk C ( i , j ) = C k - e - h

k = 2 k !

X k = k - e - i - h e - h k = O k!

= X(1- e c X ) .

The average contention for all nodes is then

Since X = rF, this expression depends on the distribution F( i, j ) . Unfortunately, it is difficult to find a closed-form expression for F under the greedy policy. It is not difficult, however, to write a computer program that computes this expression exactly for small factory sizes. For an 8 x 8 factory, the results are as follows:

0.0098 0.0186 0.0255 0.0296 0.0296 0.0255 0.0186 0.0098

0.0098 0.0186 0.0255 0.0296 0.0296 0.0255 0.0186 0.0098

0.0093 0.0175 0.0231 0.0255 0.0255 0.0231 0.0175 0.0093

0.0081 0.0144 0.0175 0.0186 0.0186 0.0175 0.0144 0.0081

0.0052 0.0081 0.0093 0.0098 0.0098 0.0093 0.0081 0.0052

( 3 )

The speed for one AGV is then Sln2 or

4n(n2- 1) 3

U =

Many AGV's: Let F(i , j ) be the frequency with which node (i, j ) is occupied if there is only 1 AGV in the factory. As long as

1 max F(i , j ) < <-

l J r

It can be seen that the greedy policy results in a very nonuniform occupancy frequency. About 42 percent of the time the AGV is at one of the central 25 percent of the nodes, and more than 71 percent of the time it is at one of the central 50 percent of the nodes. This nonuniformity clearly increases the contention (2.

Plausible Lower Bound on Multiple AGV Speed

assume that there is a policy with three properties: In order to establish a plausible lower bound on speed,

Minimal path length: The paths are the same length as

Minimal contention: All the F(i , j ) are equal to l /n2. Minimal delay: Whenever AGV's are in contention, the

those of the greedy policy.

GROSSMAN: TRAFFIC CONTROL OF ROBOT VEHICLES 495

resolution of this contention requires each AGV to lose one time step, with a plausible additional factor that is inversely proportional to the probability of adjacent nodes being available for evasive action.

Actually, such a policy is not feasible, but it is a useful assumption to get a lower bound on AGV speed. In this situation,

where

r n2

A = - .

When there is contention among k vehicles, it is plausible to say that the evasive action costs each AGV one step, provided that all adjacent nodes are empty. But as r increases, the adjacent nodes are likely to be occupied. The probability of each node being unoccupied is e - Thus it is plausible to say that the evasive action actually costs each vehicle eh steps. The average number of steps that each AGV spends waiting is therefore SrCeh. Thus

S, - S, Ceh = SI

where SI , the number of steps in the absence of contention, was derived in (3). Therefore,

4(n2- 1 ) 3 n [ l -h (eh- l ) ]

S , =

and a lower bound to the speed is

4 ( n 2 - 1 ) 3nh[l - h ( e x - l ) ] . U =

For arbitrary r and n, this new expression provides a lower bound on U, subject to an earlier assumption that contention is a small effect. This assumption limits the validity of this equation to regions in which r < < n2.

Setting d u / A = 0 and solving numerically gives X = 0.4876. Thus the optimal number of vehicles is r = 0.4876 n2. Although this value is not really in the range for which the assumptions are completely valid, it is indicative that the AGV speeds are optimized when the number of “carriers” is approximately equal to the number of “holes,” i.e., the number of nodes with AGV’s is the same as the number without AGV’s.

This result is intuitively reasonable since the limiting cases of 0 AGV’s and n AGV’s would both result in zero throughput. This property suggests a rough symmetry between carriers and holes, from which it follows that the optimal should be near r = n2/2 .

With n2/2 AGV’s, the lower bound on speed becomes

16(n2- 1 )

3n(3 - &> *

U =

This equation indicates that as the factory is made arbitrarily

I l l : ! V m i d p t 1 I I I +-------+-------+----TARGET3----+-------+

Fig. 5. Examples of counterclockwise flow for targets in Quadrants 2 and 3.

large, the AGV speed must scale as n. Comparing this equation with (2) shows that having independent AGV’s with autonomous route selection is preferable to restricted road- ways, for which the AGV speed scales as n2.

Even if most of the mathematics of this paper is disre- garded, the different scaling of speed AGV follows directly from the fact that with roadways the average path length scales as n 2 , while with autonomous routing it scales as n. Even though the number of AGV’s scales with the number of machine tools, the speed of each must scale with path length in order to complete each task in the same time.

Benevolent Counterclockwise Flow Policy It remains to demonstrate the existence of a simple feasible

traffic policy with performance close to the derived lower bound. Suppose that each AGV follows the following rules:

1 ) From the AGV’s own ( i , j ) location, determine in which quadrant q the target node ( i ’ , j’) lies:

Quadrant 1 has i’ > i and j ’ L j . Quadrant 2 has j‘ > j and i’ I i . Quadrant 3 has i’ < i and j’ 5 j. Quadrant 4 has j’ < j and i’ 2 i .

2) Depending on the value of q, try to move to an adjacent

If q is 1 then ( i + 1 , j). If q is 2 then ( i , j + 1 ) . If q is 3 then ( i - 1 , j ) . If q is 4 then ( i , j - 1 ) .

3) If that node is blocked, add 1 to q and try Step 2 again. 4 ) If that node is blocked, add 1 to q and try Step 2 again. 5 ) If that node is blocked, add 1 to q and try Step 2 again. 6) If all adjacent nodes are blocked, then wait at the current

If there is only one AGV, these rules cause it to follow 2- segment paths in a generic fylfot pattern. This pattern has the property that the AGV motion is always counterclockwise around the midpoint between the source and sink, as shown for example in Fig. 5 .

The integrity of the pattern is lost when many vehicles are in

node:

node.

496

100

10 0 IO 20 30 40 50 60

r=NUMBER OF AGV’S

Fig. 6. Simulation results.

contention, but intuitively there is still an overall counter- clockwise flow of traffic. Aditionally, the probable action when there is contention adds little to the total number of steps. For instance, suppose the AGV in this figure were heading towards target 2 but was immediately blocked from moving up. Then it would instead try to move left, which would also bring it one arc closer to target 2.

Simulation Results Fig. 6 shows four superimposed graphs for the AGV speeds

as a function of r in an 8 x 8 factory. The lowermost curve gives the speed in the absence of any traffic contention. The next curve gives the lower bound that was computed earlier. This curve is shown over a domain of r that extends into the region of large r for which the assumptions used in the derivation are no longer valid. The third and fourth curves show the behavior of the benevolent and greedy policies, respectively, as determined by extensive Monte Carlo simula- tions.

For the optimum number of AGV’s, the curves show that the speeds for the greedy policy, the benevolent policy, and the derived lower bound are 64, 51, and 32 arcs/step, respectively. These numbers indicate that the greedy policy is typically about 25 percent worse than the benevolent policy. Conversely, for the optimum number of AGV’s, the derived lower bound is apparently 38 percent better than the benevo- lent policy. It is the author’s conjecture that the benevolent policy is actually the optimal policy for autonomous AGV’s without memory or task trading.

These curves clearly indicate that traffic contention has a big effect on AGV system performance, and the choice of traffic policy can be significant. It can also be seen that for the benevolent policy, the optimal value of r is near n2/2, indicating that this intuitive value offers a reasonable guideline for optimizing the number of AGV’s.

HIGHWAYS AND SUPERHIGHWAYS

The fact that AGV speed must scale as some power of factory size, suggests that partitioning the factory recursively into minifactories connected by AGV highways and super- highways might offer some advantage.

Consider, for example, the partitioned factory shown in Fig. 7. It has 25 segments, each with t = 9 machine tools.

IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4, NO. 5 , OCTOBER 1988

/ / / /

+--+--+

+--+--+ I l l I l l +--+--+

/

+--+--+ I l l I l l +--+--+ +--+--+

/

+--+--+ I l l I l l +--+--+ +--+--+

/

+--+--+ I l l I l l +--+--+ +--+--+

/

+--+--+ I l l I l l +--+--+ +--+--+

/

+--+--+ I l l I l l +--+--+ +--+--+

/

+--+--+ +--+--+ I l l I l l +--+--+

/

+--+--+ +--+--+ I l l I l l +--+--+

1

+- -+- -+ I l l I l l +--+--+ +--+--+

/

+--+--+ I l l I l l +--+--+

+--+--+ /

+--+--+ +--+--+ I l l I l l +--+--+

/

+--+--+ +--+--+ +--+--+

I l l I l l

/

+--+--+

+--+--+ +--+--+

I I ! I l l

/

Fig. 7. Partitioned factory with highways.

Within the segments there are conventional AGV roads; between the segments there are highways. AGV’s may transfer between the ordinary roads and highways only at the nodes of the highway system.

Two assumptions are made that accentuate the benefit of highways: First, suppose that all of the AGV’s travel at a speed of 1 arc per time step, independent of the physical length of the arcs. Second, assume that zero time is associated with transferring between roads and highways.

If the number of partitions is large, the probability for each task that both the source and sink will fall within the same partition is small. The average number of steps to bring a part from the source to the transfer corner of that partition is t( t + 1). This cost is incurred 4 times for each task. On the highway, the average number of steps per task is given by (3). The average number of steps per task for a single AGV is therefore

3 - f n \ \ t /

This expression is minimized when

6nt3 + (3n - l ) t z - n2 = 0.

When n is very large, the second term may be neglected, and

As a result, S ; scales as n2’3, whereas without partitioning SI scales as n. Thus for a solitary AGV, highways can reduce the number of steps, if the size of the factory is very large.

GROSSMAN: TRAFFIC CONTROL OF ROBOT VEHICLES 497

On the other hand, as the number of AGV’s is increased, there would be traffic jams at the transfer points for each partition. Also, it should be noted that the value given by (5) is quite small. For instance, if n is 48, then t is only 2. This result is in spite of the assumptions made to accentuate the benefits. The implication is that for factories of realistic size, the benefits of partitioning would be very small, if they exist at all. For this reason, an analysis of the situation with multiple AGV’s will not be given.

Superficially, the limited benefit in partitioning factories seems to conflict with the growing practice of grouping machine tools into small localized sets whose capabilities match particular sets of parts to be manufactured. The classification of parts to make this grouping possible is called Group Technology (GT) [ 131. It is significant, however, that in the GT situation, the probability distributions for sources and sinks are no longer independent, as assumed in all of the earlier analysis. Clearly, when there is a high conditional probability of sinks being near sources, then there is much greater benefit in partitioning the factory and providing highways and superhighways between partitions.

In this context, a more general problem would be to consider the entire range of parts to be manufactured and then use the derived conditional probabilities of moving between pairs of machine tool types in order to optimize the layout of the entire factory. Similar problems have been treated heuristically in the literature [ 121. These problems are, however, outside the scope of the current paper, which assumes that all machine tools have equal and independent probabilities, in which case there can be no issue of optimizing the layout.

The more general problem is similar to problems addressed in the technical literature on the optimization of computer resources. The relevant problems fall into two categories: storage allocation and networking. For storage allocation, the literature typically deals with a one-dimensional problem [8] instead of a two-dimensional factory floor. For networking, the relevant work involves both geographically distributed systems and multiprocessor supercomputer systems [ 11. In both cases, the space is usually much more than two- dimensional, and the metric is not physical distance, as it is for a factory without highways.

CONCLUSION This paper has addressed the control of traffic when there

are large numbers of Automatic Guided Vehicles (AGV’s) in a factory. It has shown that traffic contention can have a major effect on the collective performance of the AGV’s The paper analyzed in detail two policies for managing autonomous independent AGV’s in a grid-iron network of roads and

showed that one of these policies is close to optimal. It also discussed under what conditions improvement can be obtained by modifying the factory to have highways and superhigh- ways.

ACKNOWLEDGMENT

The motivation for this paper came directly from comments made by Y. Abe of Mazda Corporation. M. Lavin suggested the similarities of this problem to those of computer resource allocation. L. Nackman made helpful editorial suggestions.

REFERENCES D. P. Agrawal, V. K. Janakiram, and G. C. Pathak, “Evaluating the performance of multicomputer configurations,” Computer, vol. 19, no. 5, pp. 23-37, May 1986. J. J. Bartholdi, L. K. Platzman, R. L. Collins, and W. H. Warden, “A minimal technology routing system for meals on wheels,” Interfaces, vol. 13, pp. 1-8, June 1983. P. P. Bose, “Basics of AGV systems,” Amer. Mach. Automat. Manuf., vol. 130, no. 3, pp. 105-122, Mar. 1986. R. Chattergy, “Some heuristics for the navigation of a robot,” Int. J. Robotics Res., vol. 4, no. 1, pp. 59-66, ‘Spring 1985. W. Feller, An Introduction to Probability Theory and Its Applica- tions. New York, NY: Wiley, 1950. R. E. Fenton, “On future traffic control: Advanced systems hard- ware,” IEEE Trans. Vehicular Technol., vol. VT-29, no. 2, pp. 200-207, May 1980. D. C. Gazis, Ed., Traffic Science. New York, NY: Wiley-lnters- cience, 1974. D. D. Grossman and H. Silverman, “Placement of records on a secondary storage device to minimize access time,” J. ACM, vol. 20, no. 3, pp. 429-438, July 1973. S. K. Kambhampati and L. S . Davis, “Multiresolution path planning for mobile robots,” IEEE J. Robotics Automat., vol. RA-2, no. 3, pp. 135-145, Sept. 1986. W. L. Maxwell and J. A. Muckstadt, “Design of automatic guided vehicle systems,” IIE Trans., vol. 14, no. 2, pp. 114-124, June 1982. D. Newton, “Simulation model calculates how many automated guided vehicles are needed,” Indust. Eng., vol. 17, no. 2, pp. 68-78, Feb. 1985. M. J. Rosenblatt, “The dynamics of plant layout,” Manag. Sci., vol. 32, no. 1, pp. 76-86, Jan. 1986. D. L. Shunk, “Group technology provides organized approach to realizing benefits of CIMS,” Ind. Eng., vol. 17, no. 4, pp. 74-80, Apr. 1985. P. A. Steenbrink, Optimization of Transport Networks. London, UK: Wiley-lnterscience, 1974.

David D. Grossman (M’83-SM’86) received the B.A. and Ph.D. degrees in physics from Harvard University, Cambridge, MA.

After sepending three years as an instructor at Princeton University, Princeton, NJ, he joined IBM Research, Yorktown Heights, NY, in 1970. At IBM he has been involved in robotics and manufacturing research for the past sixteen years, coauthoring over 50 technical papers and managing the Automation Research Department for about eight years. He is currently on sabbatical at the Stanford Center for

Integrated Systems, Palo Alto, CA.