Establishing a mobile conference call under delay and bandwidth constraints

Abstract

Abstract-The issue of tracking a group of users is discussed in this study. Given the condition that the search is over only after all the users in the group are found, this problem is called the Conference Call Search (CCS) problem. The goal is to design efficient CCS strategies under delay and bandwidth constraints. While the problem of tracking a single user has been addressed by many studies, to the best of our knowledge, this study is one of the first attempts to reduce the search cost for multiple users. Moreover, as oppose to the single user tracking, for which one can always reduce the expected search delay by increasing the expected search cost, for a multiple users search the dependency between the delay and the search cost is more complicated, as demonstrated in this study. We identify the key factors affecting the search efficiency, and the dependency between them and the search delay. Our analysis shows that under tight bandwidth constraints, the CCS problem is NP-hard. We therefore propose a search method that is not optimal, but has a low computational complexity. In addition, the proposed strategy yields a low search delay as well as a low search cost. The performance of the proposed search strategy is superior to the implementation of an optimal single user search on a group of users.

The conference call search problem in wireless networks

“…This last result extends an earlier result of Bar-Noy and Naor [3] for d = 2. We also show that if d is a part of the input, then finding an optimal oblivious search protocol of a tight instance becomes NP-hard in the strong sense.…”

“…We recall that Bar-Noy and Naor [3] proved that finding the optimal oblivious search protocol is NP-hard for B = 1 and d = m = 2. In this section we extend this result to the general tight case.…”

“…Proof. The claim for d = m = 2 is proved in [3]. We prove the claim for d 3 using a reduction from the PARTITION problem (see problem SP12 on p. 223 in [5]).…”

“…where (2) follows by Lemma 4,and (3),(5) follow by simple algebra. Next, (4) follows since given a set of m independent random events, the probability of their union is multiplied by at most (1 + 3 ) if we multiply the probability of each event in this set by that amount, and if we increase the probability of each event by an additive factor of = m 2 , then the probability of the union increases by at most m = m 2 2 .…”

“…The paper [3] introduced the model where search requests for different users for the same cell are made separately (i.e., we cannot ask a cell what is the subset of the users that are currently connected to it). They showed that the case B = 1, d = m = 2, is NP-hard for oblivious search protocols.…”