FIFO is Unstable at Arbitrarily Low Rates ( Even in Planar Networks ) ∗

We prove that the FIFO (First-In-First-Out) protocol is unstable in the standard model of Adversarial Queueing Theory [7] for arbitrarily low rates of packet injection. In order to prove this, we proceed as follows: (1) We first consider the extension of the standard model to networks with dynamic capacities, which was introduced in [8]. We assume that each network link may arbitrarily take on a value in the two-valued integer set {1, C} where C > 4 is an integer parameter (the high capacity). Here, for any r > 0, we construct a FIFO network (whose size is a small polynomial in 1 r ) which is unstable at any rate at least r in this setting. (2) Then, we show how to simulate the construction in (1) in order to produce a FIFO network with all link capacities being now equal to C, which is also unstable at any rate at least r in this setting. (3) Finally, we provide a simple simulation of the construction in (2) in order to produce a FIFO network (whose size is still a small polynomial in 1 r ) with all capacities being now equal to 1, which is similarly unstable. Since all capacities are equal to 1 in the standard model of Adversarial Queueing Theory [7], this implies our main result: FIFO is unstable in the standard model of Adversarial Queueing Theory model for arbitrarily low rates of packet injection. We emphasize that all of our networks are planar; we allow though the paths of packets to have cycles of edges that can be repeated a bounded number of times. Our result closes a major open problem, that of FIFO (in)stability, in the standard model of Adversarial Queueing Theory, which was already posed in the original pioneering work of Borodin et al. [7]. Note: Due to lack of space, many of our proofs are only sketched in this extended abstract; full proofs are included in a clearly marked Appendix that may be read at the discretion of the Program Committee. This work has been accepted for (electronic) publication in Electronic Colloquium on Computational Complexity (ECCC), ECCC Reports 2003, Technical Report TR03-016, accepted on March 24, 2003. This work has been partially supported by the IST Program of the European Union under contract numbers IST-1999-14186 (ALCOM-FT) and IST-2001-33116 (FLAGS), by funds from the Joint Program of Scientific and Technological Collaboration between Greece and Cyprus, and by funds for the promotion of research at University of Cyprus. † Contact Author. Department of Computer Engineering and Informatics, University of Patras, Rion, 265 00 Patras, Greece, & Computer Technology Institute, 61 Riga Feraiou, P. O. Box 1122, 261 10 Patras, Achaia, Greece. Fax: +30261-0-960442, Email: Dimitrios.Koukopoulos@cti.gr Departemnt of Computer Science, University of Cyprus, 1678 Nicosia, Cyprus. Email: mavronic@ucy.ac.cy Department of Computer Engineering and Informatics, University of Patras, Rion, 265 00 Patras, Greece, & Computer Technology Institute, P. O. Box 1122, 261 10 Patras, Greece. Email: spirakis@cti.gr Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 16 (2003)