Paging Mobile Users Efficiently and Optimally

A mobile user is roaming in a zone composed of N cells in a cellular network system. When a call to the mobile user arrives, the system pages the mobile user in these cells since it never reports its location unless it leaves the zone. The N cells are associated with a probability vector (p₁, ...,p_N) where p_i is the probability that the mobile user resides in the ith cell and all the probabilities are independent. A delay constraint paging strategy must find the mobile user within D (1 les D les N) paging rounds; in each round a subset of the N cells is paged. The goal is to minimize the expected number of paged cells until the mobile user is found. Solutions based on dynamic programming that yield optimal strategies are known. The running time of the known implementations is Theta(N²D). Our first contribution is to improve the running time to Theta(ND) by proving that the dynamic programming recursive formulation satisfies the Monge property, permitting us to use various dynamic programming speedup techniques. A Theta(N) heuristic solution is also known. Our second contribution is a heuristic whose running time is Theta(N log D). Our heuristic outperforms the known heuristic while running faster for D << N. We compare the non-optimal heuristics with the optimal solution demonstrating the tradeoff between optimality and running time efficiency of various solutions.