Tandem Queue Blocking

Two processing stages are arranged in series: Stage 1 feeds work into a bounded buffer, which feeds Stage 2. Both stages have the same mean service rate $\mu$, and the arrival rate $\lambda < \mu$ so neither stage is overloaded on average. However, Stage 1 has high variance (hyperexponential service); Stage 2 has zero variance (deterministic service). Even though both stages have identical mean throughput and the system is underloaded, Stage 2 sits idle for a substantial fraction of time when the buffer between them is small. The idle fraction only vanishes as the buffer size $K \to \infty$. High service-time variance at Stage 1 produces bursts of output—many jobs finish close together—followed by droughts. With a small buffer, the burst overflows (blocking Stage 1) and the drought starves Stage 2. Both effects reduce system throughput below what we would intuitively expect. Analysis For a two-stage tandem queue with a finite buffer of capacity $K$, the blocking probability at Stage 1…