The conference call search problem in wireless networks

Cellular telephony systems, where locations of mobile users are unknown at some times, are becoming more common. Mobile users are roaming in a zone. A user reports its location only if it leaves the zone entirely. The Conference Call Search problem (CCS) deals with tracking a set of mobile users in order to establish a call. To find a single roaming user, the system may need to search each cell where the user may be located. The goal is to identify the location of all users, within bounded time, satisfying some additional constraints on the search scheme. We consider cellular systems with n cells and m mobile users (cellular phones). The uncertain location of users is given by m probability distribution vectors. Whenever the system needs to find the users, it conducts a search operation lasting at most d rounds. A request for a single search step specifies a user and a cell. In this search step, the cell is asked whether the given user is located there. In each round the system may perform an arbitrary number of such requests. An integer number B ≥ 1 bounds the number of distinct requests per cell in every round. The bounds d and B result from quality of service considerations. Every search step consumes expensive wireless links, which motivates search techniques minimizing the expected number of requests thus reducing the total search costs. We distinguish between oblivious, semi-adaptive and adaptive search protocols. An oblivious search protocol decides on all requests in advance, and stops only when all users are found. A semi-adaptive search protocol decides on all the requests in advance, but it stops searching for a user once it is found. An adaptive search protocol stops searching for a user once it has been found (and its search strategy may depend on the subsets of users that were found in each previous round). We establish the difference between those three search models. We show that for oblivious "single query per cell" systems (B = 1), and a tight environment (d = m), it is NP-hard to compute an optimal solution (the case d = m = 2 was proven to be NP-hard already by Bar-Noy and Naor) and we develop a PTAS for these cases (for fixed values of d = m). However, we show that semi-adaptive systems allow polynomial time algorithms. This last result also shows that the case B = 1 and d = m = 2 is polynomially solvable also for adaptive search systems, answering an open question of Bar-Noy and Naor.