On functional limit theorems for the cumulative times in alternating renewal processes
Introduction
Consider an alternating renewal process with i.i.d. alternating “on–off” cycles , where the and are “on” and “off” durations in the th cycle, . Assume that the process starts at the beginning of an “on” period. Let and , and and . Let for and . Then for . Define the indicator process by for each . When , the process is in the “on” period and otherwise the process is in the “off” period. Define the cumulative “on” and “off” processes and , respectively, by We focus on the analysis of the cumulative “on” time process . It is well known (see, e.g., Example 3.6(A) of Section 3.6 in [18]) that and , where
Representation of the cumulative “on” time as the first passage time. The process can be represented as the first passage time for the random walk associated with the “on–off” cycle times, as observed in [23]. Let be defined by with . Define the compound process by Then we can write directly as for . Now define an auxiliary process by Thus, we obtain the following representation of the process as the first passage time of the process :
In this paper, we first review two functional central limit theorems (FCLTs) for the cumulative “on” time process and provide new proofs for these FCLTs (Theorem 2.1, Theorem 2.2 in Section 2). In Theorem 2.1, the “on” and “off” times are of the same order and the result is stated in [21, Theorem 8.3.1] (ours is a slight modification) and its proof is given in Section 5.3 of [22]. That proof applies Theorem 12.5.1 (iv) of [21] by controlling the oscillations of the cumulative “on” time process in the Skorohod topology. Our new proof takes advantage of the first passage time representation in (1.6) and thus applies the continuous mapping theorem for the inverse mapping with centering [21, Theorem 13.7.2]. This result has been used in establishing FCLTs for the queues with “on–off” sources (see, e.g., [20] and a good review in Section 8 of [21]).
In Theorem 2.2, the “on” and “off” times are of different orders, in particular, the “off” times are asymptotically negligible comparing with the “on” times. The result has been used in queueing systems with service interruptions and server vacations for single-server queues and networks [3], [11], [12], [21]. A similar result is also used for many-server queueing systems with service interruptions [15], [14], [16], [17]. This theorem can be proved with the argument as in the proof of Theorem 14.7.3 in [21]. The proof can also be done with an explicit construction of the parametric representations for the Skorohod topology (see Section 5.4 in [16]). Here we provide a new proof by applying the continuous mapping theorem to the inverse mapping with centering using the representation in (1.6). The new proofs for these two FCLTs for the cumulative “on” time processes provide important insights on their understanding and future applications.
We prove the strong approximations for the cumulative “on” time processes (Theorem 3.1). Although strong approximations for renewal processes have been well studied and applied in queueing theory [1], [4], [5], [6], [8], [9], [10], [19], strong approximations for the cumulative “on” time processes in alternating renewal processes have remained open in the literature. The first-passage-time representation of the cumulative “on” time process in (1.6) plays a key role in establishing the strong approximations, since some existing results and proof techniques in [4], [5] on renewal processes and the inverse mapping can be applied and/or adapted for our purpose. In Theorem 3.1, we obtain the probability bounds and almost sure properties under the condition of either finite moment generating functions of the “on” and “off” times in a neighborhood of zero or their finite moments of order higher than two.
As an application, the strong approximations of the cumulative “on” time process are applied to a single-server fluid queue with “on–off” sources. Under the proper assumptions on the strong approximations of the input processes, we obtain the strong approximations of the queueing process by a reflected Brownian motion in the critically loaded regime or a Brownian motion in the overloaded regime (Theorem 3.2). Heavy-traffic approximations for fluid queues with “on–off” sources have been well studied in the literature (see a good review in Section 8 of [21]). However, strong approximations for fluid queues with “on–off” sources have remained open. To the best of our knowledge, Theorem 3.2 is the first result on this subject.
We use (and ), , to denote real-valued -dimensional (nonnegative) vectors, and write and for . Let denote the natural numbers. For , , , and . Let denote the -valued function space of all right continuous functions on with left limits everywhere in . Denote . Let and denote the space equipped with Skorohod and topology, respectively. Let be the -fold product of with the product topology. Similarly, let be the -fold product of with the product topology. Notations and mean convergence of real numbers and convergence in distributions. Let “” denote “equal in distribution” and “” be “definition by equation”. The abbreviation a.s. means almost surely. All random variables and processes are defined on a common probability space .
Section snippets
Functional central limit theorems
In this section we state the two FCLTs for the diffusion-scaled processes of and provide new proofs for them. We index the quantities and processes with and use as a scaling parameter, and let .
Strong approximations
In this section we prove strong approximations for the cumulative “on” time process by applying the first-passage-time representation in (1.6), and then apply them to a single-server fluid queue with “on–off” sources. We make the following assumptions on the “on” and “off” times. Assumption 3 The “on” and “off” times and are independent for each . In addition, either of the following conditions holds: The moment generating functions of and satisfy and in a neighborhood of
Proofs of Theorems 3.1 and 3.2
Proof of Theorem 3.1 Let be the “extended” renewal process defined by , . Then by Theorem B in [4], there exists a standard Wiener process such that under Assumption 3(a), , for all and as , a.s., and under Assumption 3(b), , for all and as ,
Acknowledgments
G. Pang and Y. Zhou are supported in part by NSF CMMI-1538149 and J. Yang is supported in part by NSFC 11541006. The authors thank the reviewer for helpful comments.
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