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UNIVERSITY OF SOUTHERN CALIFORNIA, SPRING 2008 1 Lecture Notes 2 EE 549 — Queueing Theory Instructor: Michael Neely I. M ULTISERVER S YSTEMS AND P ERFORMANCE T RACKING Here we treat multi-server systems that share a common buffer. We begin with the following simple example. A. Example Compare a single server, work conserving queue with constant server rate μ = 2 to a system of two parallel servers with individual service rates μ = 1 . Consider an input stream consisting of exactly two packets, both of length equal to 1 unit: Packet A arrives at time t = 1 , and packet B arrives at time t = 1 . 5 . The packets are served in FIFO order in the single server system. The packets are also served in FIFO order in the 2-server system: Packet A begins its service in either of the rate- 1 servers immediately upon arrival, and packet B begins its service immediately upon its arrival by entering the alternative rate- 1 server (note that the first server is busy with packet A at this time). 1 1.5 2 2.5 3 1.5 1 0.5 O t rate=2 rate=1 rate=1 First packet finished, and 2 nd packet half finished =U multi (t) =U single (t) Fig. 1. An example of the unfinished work functions associated with two arriving packets entering a multi-server system versus entering a single-server system, illustrating the multiplexing inequality. The above figure illustrates the corresponding unfinished work functions U single ( t ) and U multi ( t ) . In this example we observe that U single ( t ) U multi ( t ) for all time t . This can be considered as a special case of the multiplexing inequality by treating the multi-queue input functions X 1 ( t ) and X 2 ( t ) as the streams consisting of the packet entering the first server and the second server, respectively. It is clear from the above example that the multi-server system suffers from inefficiencies when it is not fully loaded , that is, when there are some servers that sit idle. In this lecture we describe the worst case backlog increase incurred by these inefficiencies. B. Multi-Server, Single Buffer Queues A multi-server, single buffer queue is a queueing system with a single shared buffer for storing incoming packets, together with a set of servers that process these packets (see Figure below). A packet waiting in the queue can be processed by any one of the servers. This system is conceptually similar to a system of parallel queues, with the exception that parallel queues often have separate storage buffers. Indeed, without a shared buffer, it is difficult or impossible for a packet currently contained in one queue to switch to the buffer space of another queue that it prefers. Shared buffering allows such freedom. Definition 1: A multi-server, single buffer queueing system is work conserving if it never holds a packet in its buffer while there is an idle server available.
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UNIVERSITY OF SOUTHERN CALIFORNIA, SPRING 2008 2 X (t) K 2 μ 1 ( t 29 μ 2 ( t 29 μ Κ ( t 29 1 Fig. 2. A multi-server, shared buffer queue.
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