A Note on Applying Stochastic Network Calculus

This note is to provide results for and give hints on applying stochastic network calculus to service guarantee analysis say of wireless networks. At this moment of writing, the focus is on single hop networks. At the end of the note, potential research topics are discussed. 1 Preliminaries 1.1 Model and Notation We consider discrete-time single cell wireless networks. All nodes share a wireless channel that is stochastic in nature. These nodes may be classified, e.g. in cognitive radio networks. For ease of exposition, we assume there is only one (aggregate) flow generated by each node, and for this flow, there is a first-in-first-out (FIFO) buffer that, if not specified, is assumed to have infinite queue space. All queues are empty at time 0. In modeling a network, several processes are used. A process is defined to be a function of time t(≥ 0). It could count the (cumulative) amount of traffic (in number of bits) arriving at some network element, the amount of traffic (in number of bits) departing from the network element, the amount of service (in number of bits) provided by the network element, or the amount of service (in number of bits) that failed to be provided by the network element due to some impairment to it. Respectively, we call the process (cumulative) arrival process, denoted by A(t), (cumulative) departure process, denoted by A(t), (cumulative) service process, S(t), or (cumulative) impairment process, I(t). All such processes are defined on t ≥ 0 and by convention have zero value at t = 0. These processes are called space domain processes. Also in this note, another type of processes, called time domain processes, is also adopted in the modeling. Specifically, a process could represent the sequence of arrival times to a network element, the sequence of departure times from the network element, and the sequence of service times provided by the network element to the corresponding sequence of arrivals. There processes are respectively called the time-domain arrival process, denoted by {ai}; departure process, denoted by {di}; and service process; service process, denoted by {δ}, where i = 1, 2, . . . . Wherever necessary, we use superscripts and subscripts to distinguish different flows and different categories of network elements. However, when possible without giving confusion, the superscript or subscript will not be used. A summary of the notation is provided in Table 1. The backlog at time t is defined as B(t) = A(t) −A(t). (1) For delay, there are two definitions. The direct one is, for packet i, the total delay in the system is defined as D = d − a. (2) Another definition for delay at time t, which is sometimes called virtual delay in the literature, is as: D(t) = inf{d ≥ 0 : A(t) ≤ A(t+ d)}. (3) 1.2 Min-Plus and Max-Plus Convolutions The min-plus convolution, denoted by⊗, and the max-plus convolution, denoted by ⊗̄, of single variate functions, are respectively defined as: F ⊗G(t) = inf 0≤τ≤t {F (τ ) +G(t− τ )}, (4)