Further assume that with the scheduling policies under consideration, the system is “stable” (formally discussed later in this chapter see Section 5.6.2), and hence the following time averages exist with probability 1: N FCFS = Ñ P = N P, W FCFS and W P. By the preceding construction, it can also be easily seen that N˜ P( t) = N FCFS( t) for all t.

Because of the assumptions about the policies, a little thought shows that N P( t) and Ñs P( t) are statistically indistinguishable a probability question about either of these processes yields the same answer. Denote the multiplexer queue length (for this alternative way of sampling packet lengths) by Ñ P( t). Note that for a non-FIFO policy the kth packet to be transmitted need not be the kth packet to arrive. The packet arrival instants are unchanged, but the kth packet to be transmitted is assigned the length L k when it is scheduled for transmission. Let us now modify the system under policy P in the following way. Let P denote another policy that satisfies the preceding assumptions, and let N P( t) denote the corresponding packet queue length process. N FCFS( t) is a random process observe that it is completely determined by the random sequences a k and L k. Let N FCFS( t), t ≥ 0, denote the number of packets in the system when the scheduling policy is FCFS. The second and third assumptions ensure that when the link completes the transmission of a packet and the scheduler looks for a new packet to transmit, the service times of the waiting packets are as if they were “freshly” sampled (in an independent and identically distributed (i.i.d.) fashion) from the distribution L( l). When the transmission of a packet is initiated, the link is dedicated to the packet and the packet is transmitted completely. Thus, for example, a scheduling policy that transmits shorter packets first is not under consideration in this discussion.

Īfter the completion of the transmission of a packet, the scheduler chooses the next packet for transmission without any regard to the service times of the waiting packets.

The packet lengths ( L k, k ≥ 1) are independently and identically distributed, with some distribution L( l) that is, Pr (packet length ≤ l) = L( l). We make the following assumptions about the system: The packet arrival rate is λ (packets per second). Packets enter the multiplexer at arrival instants a k, k ≥ 1 the packet arriving at a k has the length L k.

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