Ballot theorems applied to the transient analysis of nD/D/1 queues

The problem of transporting constant-bit-rate (CBR) traffic through a packet network is analyzed. In the system considered, CBR traffic is packetized and packets from several similar sources are multiplexed on a transmission link. The bit streams are recreated at the receiving end by demultiplexing the packets and then playing out the packets of each CBR stream. Traffic fluctuations may cause gaps to appear in the playout process. Their frequency can be reduced by adding a smoothing delay to each stream. The queueing system analyzed has periodic arrivals and deterministic service times. A method of analysis, based on the ballot theorems of Takacs (1967), is presented to provide steady-state delay distributions as well as a transient analysis of the system to predict the statistics of the time for a gap to develop in the CBR stream as a function of the smoothing delay