# APPROXIMATION OF THE QUEUE LENGTH DISTRIBUTION OF GENERAL QUEUES

• Published : 1994.01.31

#### Abstract

In this paper we develop an approximation formalism on the queue length distribution for general queueing models. Our formalism is based on two steps of approximation; the first step is to find a lower bound on the exact formula, and subsequently the Chernoff upper bound technique is applied to this lower bound. We demonstrate that for the M/M/1 model our formula is equivalent to the exact solution. For the D/M/1 queue, we find an extremely tight lower bound below the exact formula. On the other hand, our approach shows a tight upper bound on the exact distribution for both the ND/D/1 and M/D/1 queues. We also consider the $M+{\Sigma}N_jD/D/1$ queue and compare our formula with other formalisms for the $M+{\Sigma}N_jD/D/1$ and M+D/D/1 queues.

#### References

1. IEEE Trans. Commun. v.COM-27 no.3 The single server queue with periodic arrival process and deterministic service time Eckberg, A.E. Jr.
2. Proc. Int. Conf. Commun.'87 Using a packet switch for circuit-switched traffic: A queueing system with periodic input traffic Karol, M.J.;Hluchyi, M.G.
3. Proc. IEEE GLOBECOM '89 Queueing analysis of continuous bit-stream transport in packet networks Bhargava, A.;Humblet, P.A.;Hluchyi, M.G.
4. IEEE/ACM Trans. Network v.1 no.1 Ballot theorems applied to the transient analysis of nD/D/1 queues Humblet, P.;Bhargava, A.;Hluchyi, M.G.
5. IEEE Trans. Commun. v.39 no.2 The superposition of periodic cell arrival stream in an ATM multiplexer Robets, J.W.;Virtamo, J.T.
6. Loss and waiting time probability approximation for general queueing, Tech. Rep. IEICE, SSE93-2 Nakagawa, K.
7. Queueing Systems;Theory, Vol. 1 Kleinrock, L.
8. Proc. IEEE GLOBECOM '92 Manage-ment of cell delay variation in ATM networks Guilemin, F.;Monin, W.