TRANSIENT SOLUTIONS FOR THE BUFFER BEHAVIOR IN STATISTICAL MULTIPLEXING

In this paper we present time-dependent (or transient) solutions for a mathematical model of statistical multiplexing. The problem is motiva ted by the need to better understand the performance of fast packet sw itching in asynchronous transfer mode (ATM), which will be adopted in the broadband ISDN. The transient solutions will be of critical value in understanding dynamic behavior of the multiplexer, and loss probabi lities at the cell (or packet) level. We use the double Laplace transf orm method, and reduce the partial differential equation that governs the multiplexer behavior to the eigenvalue problem of a matrix equatio n in the Laplace transform domain. We derive important properties of t hese eigenvalues, by extending earlier results discussed by Anick, Mit ra and Sondhi (1982) for the equilibrium solutions. A most critical st ep in our analysis is to identify sets of linear equations that unique ly determine the time-dependent probability distributions at the buffe r boundaries. These boundary conditions are in turn used to solve the general transient solutions. For the infinite buffer case, we show tha t a closed form solution is given in terms of explicitly identified ei genvalues and eigenvectors. When the buffer capacity is finite, the de termination of boundary conditions requires us to solve a matrix equat ion. We also observe that the statistical multiplexing not only achiev es the effective bandwidth gain (i.e., a multiplexing gain), but also reduces the system's packet loss probability and shorten transient per iods. We present some numerical results to illustrate our solution tec hnique. A potential application of the time-dependent solution is in t he area of preventive congestion control in a high speed network.