1 Introduction
A general
\(2\times 2\) switch is modeled by a two-server queueing system with two arrival streams. A well-studied special case of such a switch is given by the
\(2\times 2\) clocked buffered switch, where in a unit time interval each arrival stream can generate only one arrival and each server can serve only one customer; see, for example, [
1,
11,
19] and others. This switch is commonly used to model a device used in data-processing networks for routing messages from one node to another.
In this paper, we study a \(2\times 2\) switch that operates in continuous time, i.e., the arrivals are modeled by two independent compound Poisson processes. Every incoming job is of random size and it is then distributed to the two servers by a random procedure. This leads to a pair of coupled M/G/1-queues. In this model, we study the equilibrium probabilities of the resulting workload processes. In particular, we determine the asymptotic behavior of the probabilities that the workloads exceed a prespecified buffer. Hereby, we will distinguish between workload exceedance of a specific single server, both servers, or one unspecified server. As we will see, the behavior of these workload exceedance probabilities strongly depends on whether jobs are heavy-tailed or light-tailed, and we will therefore consider both cases separately.
A related model to the one we study has been introduced in [
10] where a pair of coupled queues driven by independent spectrally positive Lévy processes is introduced. The coupling procedure, however, is completely different to the switch we shall use. For this model, in [
10], the joint transform of the stationary workload distribution in terms of Wiener–Hopf factors is determined. Two parallel queues are also considered, for example, in [
23] for an M/M/2-queue where arriving customers simultaneously place two demands handled independently by two servers. We refer to [
2,
22] and references therein for more general information on Lévy-driven queueing systems.
As it is well-known, there are several connections between queueing and risk models. In particular, the workload (or waiting time) in an M/G/1 queue with compound Poisson input is related to the ruin probability in the prominent Cramér–Lundberg risk model, in which the arrival process of claims is defined to be just the same compound Poisson process; see, for example, [
2] or [
31]. To be more precise, let
$$\begin{aligned} R(t)=u+ct-\sum _{i=1}^{N(t)} X_i, \quad t\ge 0, \end{aligned}$$
be a
Cramér–Lundberg risk process with initial capital
\(u>0\), premium rate
\(c>0\), i.i.d. claims
\(\{X_i,i\in {\mathbb {N}}\}\) with cdf
F such that
\(X_1>0\) a.s. and
\({\mathbb {E}}[X_1]=\mu <\infty \), and a claim number process
\((N(t))_{t\ge 0}\) which is a Poisson process with rate
\(\lambda >0\). Then, it is well-known that the ruin probability
$$\begin{aligned} \Psi (u)={\mathbb {P}}(R(t)<0 \quad \text {for some }t\ge 0) \end{aligned}$$
tends to 0 as
\(u\rightarrow \infty \), as long as the
net-profit condition \(\lambda \mu <c\) holds, while otherwise
\(\Psi (u)\equiv 1\). In particular, if the claims sizes are light-tailed in the sense that an adjustment coefficient
\(\kappa >0\) exists, i.e.,
$$\begin{aligned} \exists \kappa >0: \quad \int _0^\infty \hbox {e}^{\kappa x} {\overline{F}}(x) \mathop {}\!\mathrm {d}x = \frac{c}{\lambda }, \end{aligned}$$
where
\({\overline{F}}(x)=1-F(x)\) is the tail function of the claim sizes, then the ruin probability
\(\Psi (u)\) satisfies the famous
Cramér–Lundberg inequality (cf. [
2, Eq. XIII (5.2)], [
3, Eq. I.(4.7)])
$$\begin{aligned} \Psi (u)\le \hbox {e}^{-\kappa u}, \quad u >0. \end{aligned}$$
Furthermore, in this case the
Cramér–Lundberg approximation states that (cf. [
2, Thm. XIII.5.2], [
3, Eq. I.(4.3)])
$$\begin{aligned} \lim _{u\rightarrow \infty } \hbox {e}^{\kappa u}\Psi (u)=C, \end{aligned}$$
for some known constant
\(C\ge 0\) depending on the chosen parameters of the model. On the contrary, for heavy-tailed claims with a subexponential integrated tail function
\(\frac{1}{\mu } \int _0^x {\overline{F}}(y) \mathop {}\!\mathrm {d}y\) it is known that (cf. [
3, Thm. X.2.1])
$$\begin{aligned} \lim _{u\rightarrow \infty } \left( \frac{1}{\mu } \int _u^\infty {\overline{F}}(y) \mathop {}\!\mathrm {d}y \right) ^{-1}\Psi (u) =\frac{\lambda \mu }{c-\lambda \mu }, \end{aligned}$$
and in the special case of tail functions that are regularly varying this directly implies that the ruin probability decreases polynomially.
Via the mentioned duality, these results can easily be translated into corresponding results on the workload exceedance probability of an M/G/1-queue.
In this paper, we shall use an analogous duality between queueing and risk models in a multi-dimensional setting, as introduced in [
7]. This allows us to obtain results on the workload exceedance probabilities of the
\(2\times 2\) switch by studying the corresponding ruin probabilities in the two-dimensional dual risk model.
Bivariate risk models are a well-studied field of research. A prominent model in the literature that can be interpreted as a special case of the dual risk model in this paper has been introduced by Avram et al. [
5]. In this so-called
degenerate model, a single claim process is shared via prespecified proportions between two insurers (see, for example, [
4‐
6,
24,
26]). The model allows for a rescaling of the bivariate process that reduces the complexity to a one-dimensional ruin problem. Exact results and sharp asymptotics for this model have been obtained in [
4], where also the asymptotic behavior of ruin probabilities of a general two-dimensional Lévy model under light-tail assumptions is derived. In [
24], the degenerate model is studied in the presence of heavy tails; specifically asymptotic formulae for the finite time as well as the infinite time ruin probabilities under the assumption of subexponential claims are provided. In [
26], the degenerate model is extended by a constant interest rate. In [
6], another generalization of the degenerate model is studied that introduces a second source of claims only affecting one insurer. Our risk model defined in Sect.
2.2 can be seen as a further generalization of the model in [
6] because of the random sharing of every single claim, compare also with Sect.
5.2. There exist plenty of other papers concerning bivariate risk models of all types and several approaches to tackle the problem. For example, [
14,
21] consider bivariate risk models of Cramér–Lundberg-type with correlated claim-counting processes and derive partial integro-differential equations for infinite-time ruin and survival probabilities in these models. Various authors focus on finite time ruin probabilities under different assumptions; see, for example, [
15‐
17,
29,
32,
37,
38]. For example, in [
38], the finite time survival probability is approximated using a so-called bivariate compound binomial model and bounds for the infinite-time ruin probability are obtained using the concept of association.
In general dimensions, multivariate ruin is studied in [
9,
12,
13,
25,
30,
34]. In particular, in [
9], a bipartite network induces the dependence between the risk processes, and this model is in some sense similar to the dual risk model in this paper. Further, in [
20], multivariate risk processes with heavy-tailed claims are treated and so-called ruin regions are studied, that is, sets in
\({\mathbb {R}}^d\) which are hit by the risk process with small probability. Multivariate regularly varying claims are also assumed in [
27] and [
30], where in [
27] several lines of business are considered that can balance out ruin, while [
30] focuses exclusively on simultaneous ruin of all business lines/agents. Further, [
36] introduces a notion of multivariate subexponentiality and applies this on a multivariate risk process. Note that [
27,
36] both consider rather general regions of ruin, and some of the results from these papers will be applied on our dual risk model.
The paper is outlined as follows: In Sect.
2, we specify the random switch model that we are interested in and introduce the corresponding dual risk model. Section
3 is devoted to study both models under the assumption that jobs/claims are heavy-tailed, and it is divided into two parts. First, in Sect.
3.1, we focus on subexponentiality. As we shall rely on results from [
36], we first concentrate on the risk model in Sect.
3.1.1, and then transfer our findings to the switch model in Sect.
3.1.2. Second, we treat the special case of regular variation in Sect.
3.2, where we start with results for the risk model in Sect.
3.2.1, taking advantage of results given in [
27], before we transfer our findings to the switch model in Sect.
3.2.2. In Sect.
4, we assume all jobs/claims to be light-tailed and again first consider the risk model in Sect.
4.1 before converting the results to the switch context in Sect.
4.2. Two particular examples of the switch will then be outlined in Sect.
5, where we also compare the behavior of the exceedance probabilities for different specifications of the random switch via a short simulation study in Sect.
5.3. The final Sect.
6 collects the proofs of all our findings.
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