Waiting-time approximations for multiple-server polling systems

S.C. Borst, R.D. van der Mei
1998 Performance evaluation (Print)  
We consider multiple-server polling systems, in which each of the servers visits the queues according to its own cyclic schedule. Such systems appear to completely defy the derivation of exact waiting-time results, which motivates the search for accurate approximations. In the present paper, we derive waiting-time approximations for asymmetric systems with the exhaustive and gated service discipline. The approximations are tested for a wide range of parameter combinations. 0 1998 Elsevier
more » ... e B.V. Examples also arise in communication networks, like the underlying communication medium in the above-mentioned distributed system (cf. also Takagi [32]). Consider e.g. a local area network (LAN), consisting of a number of stations, interconnected by a transmission ring. There are various protocols known for the medium access control in a LAN with a ring architecture. One variant is the slotted ring, i.e., the ring is subdivided into time slots of the size of a single packet, circulating at constant speed. Occupying a slot corresponds to utilizing a server. Another medium access variant that may lead to multiple-server polling, is the multiple-token ring, i.e., there are multiple rings, each with a token circulating on it, representing the right of transmission on that particular ring. Holding the token corresponds to utilizing the server. Multiple-server polling systems have received remarkably little attention in the vast literature on polling systems (cf. Takagi [31] for a comprehensive survey). One of the first studies is Morris and Wang [28] in which the servers are assumed to be independent, i.e., to visit the queues independently of each other, each server according to some cyclic schedule. A very interesting phenomenon observed by Morris and Wang is the tendency for the servers to cluster if they follow identical routes, especially in heavy traffic, cf. also [25] . Numerical experiments indicate that the bunching of servers is likely to deteriorate the system performance. Obviously, the bunching of servers is alleviated if they follow different routes. Therefore, Morris and Wang advocate the use of 'dispersive' schedules to improve the system performance. Levy et al. [22] propose bang-bang policies to avoid the bunching of servers. Levy and Yechiali [23] and Kao and Narayanan [20] study a Markovian multiple-server queue, where the servers individually go on vacation when there are no waiting customers left. Mm-any and Avi-Itzhak [27] and Neuts and Lucantoni [29] analyze a Markovian multiple-server queue, where servers break down at exponential intervals and then get repaired. In [6, 19, 21, 30] , mean response time approximations are developed to analyze the performance of LANs with multiple-token rings. Mean response time approximations oriented to LANs with a multiple-slotted ring are contained in [5, 6, 24, 33] . Ajmone Marsan et al. [24] derive the mean cycle time and bounds for the mean waiting times in symmetric systems for the exhaustive, gated, and l-limited service discipline. In [l] they illustrate how Petri-net techniques may be used to study Markovian multiple-server polling systems. Browne and Weiss [ 131 is one of the few studies in which the servers are assumed to be coupled, i.e., to visit the queues together. They obtain index-type rules for determining the visit order that minimizes the mean cycle length. Browne et al. [ 1 l] examine a completely symmetric two-queue system with an infinite number of coupled servers and deterministic service times. Browne and Kella [12] consider a two-queue system with an infinite number of coupled servers, exhaustive service, and deterministic service times at one queue and general service times at the other. Borst [7] explores the class of systems that allow an exact analysis in the case of coupled servers. Van der Mei and Borst [25] show how a broad class of multiple-server polling systems may be analyzed numerically by means of the power-series algorithm (PSA). The above-mentioned studies unanimously point out that multiple-server polling systems are extraordinarily hard to analyze. Only the studies [20, 23, 27, 29] , considering single-queue systems, and [7,1 l-131, focusing on a limited class of models with coupled servers, present any exact results. To the best of the authors' knowledge, there are no exact results known for models with independent servers, apart from some meanvalue results for global performance measures like cycle times. Motivated by the mathematical intractability, we derive in the present paper waiting-time approximations for asymmetric systems with the exhaustive and gated service discipline, in which each of the servers visits the queues according to its own cyclic schedule. The remainder of the paper is organized as follows. We present a detailed model description in Section 2. In Section 3, some preliminary results are obtained for the mean interarrival times of the various servers at the various queues, which will be repeatedly used throughout the subsequent sections. In
doi:10.1016/s0166-5316(96)00063-6 fatcat:oajgguzudfeihovm4n44fmbwla