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 motivated by the need to better understand the performance of fast packet switching 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 probabilities at the cell (or packet) level. the double Laplace transform method, and reduce the partial differential equation that governs the multiplexer behavior to the eigenvalue problem of a matrix equation in the Laplace transform domain. We derive important properties of these eigenvalues, by extending earlier results discussed by Anick, Mitra and Sondhi (1982) for the equilibrium solutions. critical step in our analysis is to identify sets of linear equations that uniquely determine the time-dependent probability distributions at the buffer boundaries. These boundary conditions are in turn used to solve the general transient solutions. For the infinite buffer case, we show that a closed form solution is given in terms of explicitly identified eigenvalues and eigenvectors. When the buffer capacity is finite, the determination of boundary conditions requires us to solve a matrix equation. o observe that the statistical multiplexing not only achieves the effective bandwidth gain (i.e., a multiplexing gain), but also reduces the systemʹs packet loss probability and shorten transient periods. sent some numerical results to illustrate our solution technique. A potential application of the time-dependent solution is in the area of preventive congestion control in a high speed network.