TY - JOUR

T1 - Queue-length balance equations in multiclass multiserver queues and their generalizations

AU - Boon, M.A.A.

AU - Boxma, O.J.

AU - Kella, O.

AU - Miyazawa, M.

PY - 2017/8/1

Y1 - 2017/8/1

N2 - A classical result for the steady-state queue-length distribution of single-class queueing systems is the following: The distribution of the queue length just before an arrival epoch equals the distribution of the queue length just after a departure epoch. The constraint for this result to be valid is that arrivals, and also service completions, with probability one occur individually, i.e., not in batches. We show that it is easy to write down somewhat similar balance equations for multidimensional queue-length processes for a quite general network of multiclass multiserver queues. We formally derive those balance equations under a general framework. They are called distributional relationships and are obtained for any external arrival process and state-dependent routing as long as certain stationarity conditions are satisfied and external arrivals and service completions do not simultaneously occur. We demonstrate the use of these balance equations, in combination with PASTA, by (1) providing very simple derivations of some known results for polling systems and (2) obtaining new results for some queueing systems with priorities. We also extend the distributional relationships for a nonstationary framework.

AB - A classical result for the steady-state queue-length distribution of single-class queueing systems is the following: The distribution of the queue length just before an arrival epoch equals the distribution of the queue length just after a departure epoch. The constraint for this result to be valid is that arrivals, and also service completions, with probability one occur individually, i.e., not in batches. We show that it is easy to write down somewhat similar balance equations for multidimensional queue-length processes for a quite general network of multiclass multiserver queues. We formally derive those balance equations under a general framework. They are called distributional relationships and are obtained for any external arrival process and state-dependent routing as long as certain stationarity conditions are satisfied and external arrivals and service completions do not simultaneously occur. We demonstrate the use of these balance equations, in combination with PASTA, by (1) providing very simple derivations of some known results for polling systems and (2) obtaining new results for some queueing systems with priorities. We also extend the distributional relationships for a nonstationary framework.

KW - Balance equations

KW - Distributional relationship

KW - Nonstationary framework

KW - Palm distribution

KW - Queue length

KW - Steady-state distribution

UR - http://www.scopus.com/inward/record.url?scp=85021050492&partnerID=8YFLogxK

U2 - 10.1007/s11134-017-9528-z

DO - 10.1007/s11134-017-9528-z

M3 - Article

AN - SCOPUS:85021050492

VL - 86

SP - 277

EP - 299

JO - Queueing Systems: Theory and Applications

JF - Queueing Systems: Theory and Applications

SN - 0257-0130

IS - 3-4

ER -