Quasi-stationary Analysis of a Nite Capacity Asynchronous Multiplexer with Modulated Deterministic Input

In this paper a nite capacity multiplexer in ATM environments with modu lated deterministic sources is investigated We consider three cases of input tra c a xed number of deterministic cell generating sources a xed number of sources each modulated by an ON OFF process and a number of ON OFF sources modulated by a call arrival process in conjunction with a connection admission control scheme The main performance measures are the cell and call blocking probabilities For case we give a tight upper bound of the transient phase for the state process to reach stationary condi tions The state probabilities and the cell blocking probability are determined by means of an exact analysis For more complex modulating processes e g the cases and the compound state process and the overall cell blocking probability are analyzed using a quasi stationary approximation technique The approximation is validated by means of simulations The analysis presented is shown to be able to give reasonable approximate results to analyze the ATM multiplexer performance also for cases in which conventio nal simulative and analytic methods approach already computing time and complexity boundaries ATM multiplexer modelling and quasi stationary analysis General model We consider a multiplexer in Asynchronous Transfer Mode ATM systems which mul tiplexes the cell tra c of a number of uncorrelated and isochronous input sources e g voice tra c All input tra c sources have a cell duration and the output line has the cell duration time This leads to a speedup factor of N The multiplexer operates in discrete time so the input lines are slotted by and the output line by N During a slot length the multiplexer can transmit exactly one cell from an input line Each source is assumed to be of ON OFF type i e it will be alternatingly in two phases the cell generating phase TACT and the silence phase TSIL We consider both phases TACT and TSIL to be geometrically distributed with means and according to PrfTACT kg k PrfTSIL kg k k The number of waiting places is assumed to be nite by S For an ON OFF source a line activity factor a is de ned by a E TACT E TACT E TSIL The class of multiplexer model considered in this paper has been the subject of a number of studies Such systems with a xed number of sources have been investigated by analysis or simulation in several cases and consider a multiplexer with an in nite bu er where no blocking of cells can occur Taking into account a nite bu er have derived an analytical solution for the case where the o ered tra c does not exceed the multiplexer capacity where the blocking phenomenon is also considered Observing the input process of tra c streams o ered to the multiplexer at a slot we will see a number of sources being in the cell generating phase Clearly can be controlled by another random process which will be referred to in this paper as the modulating process Using an appropriate model to generate the desired process for a wide class of Generally Modulated Deterministic Process GMDP can be constructed In this paper we focus our interest on the calculation of the cell blocking probability assuming some particular modulating processes The following input processes will be investigated We start with a xed number const of deterministic cell generating sources where an exact calculation algorithm for the cell blocking probability will be derived This recursive algorithm is a generalization of the recursion given in bb call arrival process connection admission control ON OFF sources