Analysis of a One-Dimensional Continuous Delay-Tolerant Network Model

Abstract

The packet speed and transmission cost are examined, for a single packet traveling along a simple onedimensional, continuous-time network, using a combination of wireless transmissions and physical transports. We assume that the network consists of two nodes moving at constant speed on a circle, and changing their direction of travel after independent exponential times. The packet wishes to travel in the clockwise direction as fast and as far as possible. It travels either by being physically transported on a node's buffer, or by being wirelessly transmitted to the other node when the two are in the same location. We derive exact, explicit expressions for the long-term average packet speed (in the clockwise direction), and also for the average wireless transmission cost. These results can be viewed as initial steps towards the development of analogous exact expressions for the speed and cost, in more realistic, twodimensional wireless delay-tolerant network models.

A Simple Network of Nodes Moving on the Circle

“…φ (Φ t ∈ A) ≥ P φ (G) = P φ M + (x 0 (1), d(1), B 1 , d (1), t, ) m j=2 P φ M − (x 0 (j), d(j), B k , d (j), t, ) , and the bound in(10) and hence the result of the proposition follow from the bound in the lemma immediately below.Lemma A.2. In the notation and under the assumptions of the above proof, for any position x ∈ S, and pair of directions d, d ∈ {+1, −1}, any measurable C ⊂ S, and any initial state φ ∈ Σ, we have,P φ M + (x, d, C, d , t, ) ≥ c L (C),with c = 1 4 e −1 2 e −3t/2 .…”

A simple network of nodes moving on the circle

“…implies that,P (Φ t ∈ A) ≥ P (G) = P ( M + (x 0 (1), (1), B 1 , ′ (1), t, ) ) m ∏ j=2 P ( M − (x 0 (j), (j), B k , ′ (j), t, ) ) ,and the bound in(10) and hence the result of the proposition follow from the bound in the lemma immediately below. ▪ In the notation and under the assumptions of the above proof, for any position x ∈ S, and pair of directions , ′ ∈ {+1, −1}, any measurable C ⊂ S, and any initial state ∈ Σ, we have,P ( M + (x, , C, ′ , t, ) ) ≥ c  (C), with c = 1 4 e −1 2e −3t∕2 .…”