Search in Unknown Random Environments Erol Gelenbe ∗† Professor in the Dennis Gabor Chair Department of Electrical and Electronic Engineering, Imperial College, London SW7 2BT, United Kingdom (Dated: July 2010) N searchers are sent out by a source in order to locate a fixed object which is at a finite distance D, but the search space is infinite and D would be in general unknown. Each of the searchers has a finite random life-time, and may be subject to destruction or failures, and it moves independently of other searchers, and at intermediate locations some partial random information may be available about which way to go. When a searcher is destroyed or disabled, or when it “dies naturally”, after some time the source becomes aware of this and it sends out another searcher, which proceeds similarly to the one that it replaces. The search ends when one of the searchers finds the object being sought. We use N coupled Brownian motions to derive a closed form expression for the average search time as a function of D which will depend on the parameters of the problem: the number of searchers, the average life-time of searchers, the routing uncertainty, and the failure or destruction rate of searchers. We also examine the cost in terms of the total energy that is expended in the search. 1. INTRODUCTION It is very common to search for a known or recognis- able object, without knowing the path to the object in a precise manner, or knowing only imprecise information. Once the searcher is close to the object being sought, it can detect or recognise it (e.g. smell generated by food), however the challenge is to get close enough to it without exact information about where it is. If the search space is infinite, the searcher may also become lost “forever”. The search process may be dangerous for the searcher and it may be destroyed (e.g. eaten by a predator). The searcher will often have a finite life-span (e.g. finite fuel for a mobile robot), which is known at least in probabilis- tic terms. Examples of such situations include: (i) forag- ing by an animal that can recognize an edible object or a mate when it finds it, but does not know exacly where to find it, and can also be harmed during the search, (ii) packets traveling in a very large wireless or wired network [1, 2], without the benefit of fully reliable routing tables in intermediate network nodes, with possible packet loss due to transmission errors or buffer overflows, (iii) com- puter search of specific data or a complex digitally rep- resented object (which may be a visual entity) in a very large number data base [3], with a finite computational budget (life-span), and the possibility that the software may fail to run in some part of the database, (iv) motion of a particle under the effect of a random field; in this case the “object being sought” may be a location with an * The author dedicates this work to the memory of Professor En- rico Magenes of the University of Pavia, who spent several years during his youth in concentration camps during the Second World War, and then went on to become a leader in the field of applied mathematics and partial differential equations. † Electronic address: e.gelenbe@imperial.ac.uk opposite electric charge that ends the movement of the “searcher” particle: the finite life-span can result from the decay of the searcher’s charge, (v) motion of a bio- logical agent [4] until it docks onto a specific site where it can become active, while it loses its reactivity as it ages, (vi) motion of a physical robot sent to find a specific ob- ject with a finite reserve of fuel. For instance, packets in an ad-hoc wireless network travel over a random num- ber of relay nodes towards a destination whose location may not be precisely known [5]. In such systems once a packet reaches its destination it may be able to send back an acknowledgement by reversing the path it used and avoiding any repetitions in the nodes visited. If the source has not heard back from the packet after some predetermined time (the “time-out”), the sender sends out another packet on the assumption that the previous packet has been lost; if the current packet is not lost or dead, it will self-destroy at the same time-out to avoid having duplicate packets in the network [6, 7]. The work in the present paper is primarily motivated by (ii) and (iv) above, in that the departure point of all the searchers is exactly the same, so that their distance to the object being sought is an identical quantity D for all of the searchers. Starting from different perspectives, several authors have analysed such systems with different physical as- sumptions. For instance, the work in [8, 9] models the search behaviour of an animal which replenishes its en- ergy supply while it forages and searches; energy dissi- pation and replenishment are judiciously included in the Langevin equation used to represent the search process, and an approximate solution is obtained. In [10] the search space is represented by a sequence of finite graphs with probabilistic connection, as one may represent a wired computer network or a system of roads, and de- tailed first passage time probabilities are derived for a random initial search location. Search for a prey which