, whereis the velocity of the mobile nodes. We then describe a scheme that achieves the optimal order of delay for any given throughput. The scheme varies (i) the number of hops, (ii) the transmission range and (iii) the degree of node mobility to achieve the optimal throughput-delay trade-off. The scheme produces a range of models that capture the Gupta-Kumar model at one extreme and the Grossglauser-Tse model at the other. In the course of our work, we recover previous results of Gupta and Kumar, and Grossglauser and Tse using simpler techniques, which might be of a separate interest.

Gupta and Kumar (2000) introduced a random model to study throughput scaling in a wireless network with static nodes, and showed that the throughput per source-destination pair is Θ 1/ √ n log n . Grossglauser and Tse (2001) showed that when nodes are mobile it is possible to have a constant throughput scaling per source-destination pair.In most applications delay is also a key metric of network performance. It is expected that high throughput is achieved at the cost of high delay and that one can be improved at the cost of the other. The focus of this paper is on studying this trade-off for wireless networks in a general framework. Optimal throughput-delay scaling laws for static and mobile wireless networks are established. For static networks, it is shown that the optimal throughput-delay trade-off is given by D(n) = Θ(nT (n)), where T (n) and D(n) are the throughput and delay scaling, respectively. For mobile networks, a simple proof of the throughput scaling of Θ(1) for the Grossglauser-Tse scheme is given and the associated delay scaling is shown to be Θ(n log n). The optimal throughput-delay trade-off for mobile networks is also established. To capture physical movement in the real world, a random walk model for node mobility is assumed. It is shown that for throughput of O 1/ √ n log n , which can also be achieved in static networks, the throughput-delay trade-off is the same as in static networks, i.e., D(n) = Θ(nT (n)). Surprisingly, for almost any throughput of a higher order, the delay is shown to be Θ(n log n), which is the delay for throughput of Θ(1). Our result, thus, suggests that the use of mobility to increase throughput, even slightly, in real-world networks would necessitate an abrupt and very large increase in delay.

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Grossglauser and Tse (2001) introduced a mobile random network model where each node moves independently on a unit disk according to a stationary uniform distribution and showed that a throughput of Θ(1) is achievable. El Gamal, Mammen, Prabhakar and Shah (2004) showed that the delay associated with this throughput scales as Θ (n log n), when each node moves according to an independent random walk. In a later work, Diggavi, Grossglauser and Tse (2002) considered a random network on a sphere with a restricted mobility model, where each node moves along a randomly chosen great circle on the unit sphere. They showed that even with this one-dimensional restriction on mobility, constant throughput scaling is achievable. Thus, this particular mobility restriction does not affect the throughput scaling. This raises the question whether this mobility restriction affects the delay scaling.This paper studies the delay scaling at Θ(1) throughput for a random network with restricted mobility. First, a variant of the scheme presented by Diggavi, Grossglauser and Tse (2002) is presented and it is shown to achieve Θ(1) throughput using different (and perhaps simpler) techniques. The exact order of delay scaling for this scheme is determined, somewhat surprisingly, to be of Θ(n log n), which is the same as that without the mobility restriction. Thus, this particular mobility restriction does not affect either the maximal throughput scaling or the corresponding delay scaling of the network. This happens because under this 1-D restriction, each node is in the proximity of every other node in essentially the same manner as without this restriction.

Consider a d*n matrix A, with d