818 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007 Connectivity, Power, and Energy in a Multihop Cellular-Packet System Sayandev Mukherjee, Senior Member, IEEE, Dan Avidor, Member, IEEE, and Katherine Hartman Abstract—In this paper, we study a large network of sub- scriber stations (SSs) with certain common wireless capabilities and base stations (BSs) having direct connections to the wired infrastructure in addition to common wireless capabilities. SSs can communicate with the “outside world” only through the BSs. Connections to SSs without a direct (i.e., a single hop) wireless connection to any BS are established, if possible, through other SSs serving as wireless repeaters. The locations of the SSs and the BSs follow independent homogeneous planar Poisson processes. The propagation channels exhibit signal attenuation with distance and log–normal shadowing. We evaluate exactly the probability of an SS to have a direct wireless connection to any of the BSs and a lower bound on the t-hop (t arbitrary) outage probability of an SS. We then define the minimal hop-count routing algorithm and calculate the mean number of hops for routes connecting SSs to BSs, when a maximum hop-count constraint is imposed. We compute next the probability distribution of the transmit power under the assumption of perfect power control. We conclude by calculating a bound for the total mean transmit energy required to transfer a data packet from an SS to a BS over a minimal hop-count route and show that this energy is significantly lower than the corresponding value in a single-hop network operating at the same outage probability. Index Terms—Multihop network, outage probability, routing, transmit energy. I. I NTRODUCTION W E STUDY a large network of wireless transceivers that we call nodes. Two types of nodes are involved: sub- scriber stations (SSs) with certain common wireless capabilities and base stations (BSs) having direct wideband connection to the wired infrastructure in addition to the common wireless capabilities. SSs can communicate with the “outside world” only through the BSs. To augment connectivity, connections to SSs without a direct (i.e., a single hop) wireless connection to any BS are established, if possible, through other SSs serving as wireless repeaters, as long as the number of hops does not exceed a prescribed limit. Regular SSs are mobile or installed at customer premises, and their locations cannot be predicted ahead of time; therefore, we assume that their locations are random. We further assume that owing to practical constraints, Manuscript received January 19, 2005; revised December 1, 2005 and March 11, 2006. Parts of this work were presented at the IEEE VTC 2005- Fall, Dallas, TX, and IEEE SECON 2005, Santa Clara, CA. The review of this paper was coordinated by Dr. Q. Zhang. S. Mukherjee and D. Avidor are with Bell Laboratories, Lucent Tech- nologies, Murray Hill, NJ 07974 USA (e-mail: sayan@lucent.com; avidor@ lucent.com). K. Hartman is with the Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail: khartman@mit.edu). Digital Object Identifier 10.1109/TVT.2007.891428 availability of high-speed wired connections, and economic considerations, BSs are sparse and often cannot be positioned based on coverage considerations only. To account for this reality, we assume that the BSs, like the regular SSs, are also placed randomly over the service area. This paper focuses on two related issues. The first is the probability that an SS in a fixed arbitrary location (or while passing such a location) has a “working” wireless connection to any of the BSs, evaluated as a function of the densities of the BSs and SSs, the statistical properties of the propagation channels, and the limit set on the maximum number of hops if any. Such a limit is typically set due to delay and possibly capacity considerations. We derive analytical results and lower bounds when exact results are not obtainable. To assess the tightness of the bounds, we compare them with simulation results. We focus next on the well-known minimal hop-count routing algorithm and calculate the mean number of hops for routes connecting SSs to BSs. We then turn to the second issue, which is the probability distribution of the transmit power under the assumption of perfect power control, i.e., transmitters transmit just enough power to be “properly” received. We con- clude by calculating an expression for the total mean transmit energy required to transfer a data packet from an SS to a BS. We show that this energy is significantly lower than the corresponding value required in a single-hop network operating at the same outage probability. Multihopping can, therefore, save SS battery power. This paper is organized as follows: In Section II, we briefly discuss prior research work on connectivity in data networks. In Section III, we define the nomenclature to be used in this paper and the Poisson process that controls the spatial distribution of SSs and BSs in our system. In Section IV, we define the propagation model assumed in this paper. In Section V, we derive the distribution of the number of nodes (BSs or SSs, as the case may be) with a direct connection to an SS and an exact expression for the single-hop probability of an SS outage. We then proceed to derive a lower bound for the probability of t-hop outage. In Section VI, we describe an application where the derivations presented in the previous sections are utilized. The well-known minimal hop-count routing algorithm is pre- sented in Section VII, where we calculate the mean number of hops conditioned on a preset limit on the maximum number of hops. Section VIII considers the transmit power on multihop links in a system with perfect power control. In Section VIII-A and -B, we derive the cumulative distribution function and the mean transmit power on a single hop from an SS to a BS or another SS, respectively. In Section VIII-C, we calculate the mean total transmit energy per packet on a minimum-hop 0018-9545/$25.00 © 2007 IEEE