Ranking and candidacy probabilities of signal-to-noise ratio random variables under rayleigh fading

In design and analysis problems of wireless communication systems, a set of signal-to-noise ratio random variables may have to be dealt with. For example, this applies to transmit and/or receive diversity systems, and to scheduling and resource allocation problems. In such cases, it can be important to know how a given signal-to-noise ratio random variable ranks among the whole set of random variables. For example, in opportunistic multiuser systems, allocating resources to a user depends strongly on the probability of that user’s signal-to-noise ratio being larger than those of all (or most) other users. Considering a set of signal-to-noise ratio random variables, ranking probability is defined in this paper as the probability of a signal-to-noise ratio random variable being smaller than a given number of signal-to-noise ratio random variables in the set. Candidacy probability is defined as the probability of a signal-to-noise ratio random variable belonging to a subset of signal-to-noise ratio random variables with highest values. Closed form expressions for ranking and candidacy probabilities of signal-to-noise ratio random variables are derived, assuming Rayleigh fading. The derived probabilities are compared to those found by simulation. Comparisons confirm the correctness of the derived expressions. Simulations have been performed assuming a cellular base station that is serving a number of users. In several experiments, different levels of variation in the average signal-to-noise ratios of the users have been assumed. Results confirm the need for applying some fairness constraints in signal-to-noise ratio order-based scheduling algorithms to prevent situations where some users may not get enough access to system resources.