Hitting probabilities for compound Poisson processes in a bipartite network

Hitting probabilities for compound Poisson processes
in a bipartite network

Anita Behme  , Claudia Klüppelberg  , Gesine Reinert Technische Universität Dresden, Institut für Mathematische Stochastik, 01062 Dresden, Germany, e-mail: anita.behme@tu-dresden.deCenter for Mathematical Sciences, Technische Universität München, 85748 Garching, Boltzmannstrasse 3, Germany, e-mail: cklu@tum.deDepartment of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK, email: reinert@stats.ox.ac.uk
Abstract

This paper studies hitting probabilies of a constant barrier for single and multiple components of a multivariate compound Poisson process. The components of the process are allowed to be dependent, with the dependency structure of the components induced by a random bipartite network. In analogy with the non-network scenario, a network Pollaczek-Khintchine parameter is introduced. This random parameter, which depends on the bipartite network, is crucial for the hitting probabilities. Under certain conditions on the network and for light-tailed jump distributions we obtain Lundberg bounds and, for exponential jump distributions, exact results for the hitting probabilities. For large sparse networks, the parameter is approximated by a function of independent Poisson variables. As applications, risk balancing networks in ruin theory and load balancing networks in queuing theory are discussed.

AMS 2010 Subject Classifications:
primary:    60G51 , 60K20 , 60K25 , 91B30 ; secondary:    94C15 , 90B22

Keywords: bipartite network, Cramér-Lundberg bound, exponential jump distribution, first passage probability, hitting probability, load balancing network, multivariate compound Poisson process, multivariate ruin theory, Poisson approximation, Pollaczek-Khintchine formula, queueing theory, risk balancing network.

1 Introduction

Consider a spectrally positive compound Poisson process given by

where , is a constant, , , are i.i.d. random variables with distribution , and is a Poisson process with intensity . For such a process, the hitting probability of a given level , is denoted by and it is given by the famous Pollaczek-Khintchine formula (cf. [2, VIII (5.5)], [3, Eq. (2.2)], [14, Eq. (1.10)] or [19, Theorem 1.9])

(1.1)

whenever for . Hereby, for every distribution function of a positive random variable , we denote by for the corresponding tail, and, if has finite mean , by for the integrated tail distribution function. We denote by the -fold convolution; , such that for , .
The function of the Pollaczek-Khintchine formula (1.1) is a compound geometric distribution tail with parameter and we call the Pollaczek-Khintchine parameter.
We also recall that, whenever , it holds that for all .

In ruin theory, models the insurance risk process in the celebrated Cramér-Lundberg model with Poisson claim arrivals, premium rate and claim sizes . In this context the Pollaczek-Khintchine parameter is called the ruin parameter, and is the ruin probability for an initial risk reserve . The smaller , the smaller the ruin probability, and if , then the ruin probability as .

In queueing theory, models the workload process in a G/M/1 queue with Poisson job arrivals, service times , service rate , and a first in first out service strategy. In this context the Pollaczek-Khintchine parameter is called the traffic intensity and, if , then the workload process is stationary with stationary distribution given by , such that is the probability of an overflow of a buffer capacity . Figure III.1 on p. 46 of [3] illustrates this duality between the insurance risk process and the workload process.

In the above setting, it is well known that, when the distribution function is light-tailed in the sense that an adjustment coefficient exists; i.e.,

then the hitting probability satisfies the famous Cramér-Lundberg inequality

(1.2)

It is easy to see that a necessary condition for the existence of an adjustment coefficient is that . Further, if the are exponentially distributed with mean , then and

(1.3)

In this paper we will focus on the light-tailed case. Hence also our literature review below focuses on the light-tailed case. We derive multivariate analogues to the above classic results in a network setting. More precisely we consider a multivariate compound Poisson process whose dependency structure stems from a random bipartite network which is described in detail below. Whereas one-dimensional insurance risk processes and their analogues in the queueing setting have been extensively studied since Cramér’s introduction in the 1930s, ruin and buffer overflow for multivariate models (beyond bivariate models) are somewhat scattered in the literature; for summaries of results see [2, Ch. XII]) and [3, Ch. XIII(9)]. The duality between multivariate insurance risk processes and workload processes is proved in Lemma 1 of [6].

In an insurance risk context our framework is related to the two-dimensional setting in [4, 5] where it is assumed that two companies divide claims among each other in some prespecified proportions. Compared to these sources the main novelty of our setting is that we consider a network of interwoven companies, with emphasis on studying the effects which occur through this random network dependence structure. Our bipartite network model has already been studied in [17, 18]; there it is used to assess quantile-based measures for systemic risk, whereas in this paper we assess ruin probabilities.

In general dimensions, multivariate ruin is studied e.g. in [9, 22], where dependency between the risk processes is modeled by a Clayton dependence structure in terms of a Lévy copula, which allows for scenarios reaching from weak to very strong dependence. Further, in [11, 21], using large deviations methods, multivariate risk processes are treated and so-called ruin regions are studied, that is, sets in which are hit by the risk process with small probability.

Multivariate queueing theory is at the foundation of general analysis of stochastic processes on networks; see [16]. Using the duality between the ruin probability and the buffer overflow, [6] investigate queueing and insurance risk models with simultanous arrivals; they also give extensive references. In the context of a load balancing or loss network problem, our framework is related to [12, 15] where a user population shares a limited collection of resources. The novelty of our setting is that requests of service are partitioned and assigned randomly. This can be useful for the analysis of a network when the underlying service strategy is unknown.

The paper is structured as follows. In Section 2 we describe our bipatite network model and present the two leading examples of a complete network and a Bernoulli random network. We focus on two classical cases, namely, that a single selected component and that the sum of selected components hits a barrier. Section 3 derives results for the hitting probabilities of sums of components of the specific multivariate compound Poisson process with special emphasis on the network influence. Here we derive a network Pollaczek-Khintchine formula for component sums, a network Lundberg bound, and we present explicit results for an exponential system. In Section 4 we investigate such hitting probabilities for different network scenarios, where the now random and network-dependent Pollaczek-Khintchine parameter plays the same prominent role as does in the classical one-dimensional problem. For this parameter we derive Poisson approximations under relevant parameter settings. Section 5 provides an explicit example of a system. Section 6 is dedicated to the joint hitting probability of a set of components for which a network Lundberg bound is given and the -system is continued as an example. The final Section 7 indicates applications for risk balancing networks in ruin theory and load balancing networks in queueing theory.

2 Setting the stage

Let be a -dimensional spectrally positive compound Poisson process with independent components given by

such that for all the jump sizes are positive i.i.d. random variables having mean and distribution function satisfying . Moreover is a Poisson process with intensity , and is a constant. The corresponding constant as in the Pollaczek-Khintchine formula (1.1) will be called Pollaczek-Khintchine parameter of component .

Further we introduce a random bipartite network, independent of the multivariate compound Poisson process , that consists of agents , , and objects , , and edges between agents and object as visualized in Figure 1. If there is an edge between agent and object , we write . These links are encoded in a weighted adjacency matrix

(2.1)

for random variables , which may depend on the network and have values in such that

(2.2)

We use the degree notation

for all , . Here the variable always stand for an agent in , and the variable always stand for an object in . In case of ambiguity we add a subscript or to the degree, to read or . For a subset we abbreviate and for .

The indicators depend on the network, which is encoded in the weighted adjacency matrix ; when it clarifies the argument, we write to indicate that there is an edge between and in the network with weighted adjacency matrix .

Every object of the bipartite network is assigned to the corresponding component of the compound Poisson process . Every agent is then assigned to a resulting compound Poisson process

In total, this yields a -dimensional process of all agents given by

(2.3)

with as defined above. Hence the components of are no longer independent.

We denote by the set of all possible realizations of the weighted adjacency matrix from (2.1). Throughout, we shall denote all realisations of random quantities which are influenced by the realisations of the network structure, by the corresponding tilded letters; e.g., is a specific realisation of the process defined above.

Example 2.1.

[The homogeneous system]
A natural choice for is given by the so-called homogeneous system

i.e., every object is equally shared by all agents that are connected to it.  

Another leading example in our paper extends the one-dimensional precise hitting probability (1.3) for exponentially distributed jumps to the network setting.

Example 2.2.

[The exponential system]
We set for every agent

(2.4)

with some constant , such that the weights depend on only via some predefined subset of agents. This weighted adjacency matrix encodes that the exposure of agent group to object is inversely proportional to the expected jump size of the process associated to that object, while for fixed object with mean jump size , all which link to this object, share it in equal proportion. Here the ’s are chosen such that

The weight can be viewed as the proneness of group to possess edges. When we abbreviate .
If additionally the intensity of all compound Poisson processes does not depend on , and jumps are exponentially distributed with mean , we call the resulting model with (2.4) the exponential system. For this model, we obtain quite explicit results for the hitting probabilities depending on the network setting.  

Remarks 2.3.
  1. The independence assumption on the components of entails that jumps in different components never happen at the same time. To see that this is no mathematical restriction of the model, assume that there is dependence between the compound Poisson processes. Due to the linearity of compound Poisson processes we can disentangle the dependence through the introduction of additional compound Poisson processes. For example, if and have some dependent jumps, then let denote the process of jumps only in Object 1, the process of jumps only in Object 2, and the process of only joint jumps. Then , and are independent. Thus, mathematically, a third object, 3, is introduced, and objects 1 and 2 are altered. There is a caveat in that in the underlying network, edges to Object 1 and to Object 3, will not be independent, and the same holds for edges to Object 2 and to Object 3.

  2. We can easily extend our model to multiple layers, where e.g. the agents are connected to a set of super-agents via another bipartite network that is encoded in a second weighted adjacency matrix . The resulting process on the top layer is simply obtained by matrix multiplication in (2.3), resulting in , which reduces the problem to the form (2.3) which is treated in this paper.

  3. In an actuarial context the introduced model strongly resembles the depiction of the reinsurance market in Figure 21 of [20].

Figure 1: A bipartite network with 6 agents and 4 objects.

While many general results in this paper do not require independence of the edges in the bipartite network, our examples will always assume that the edges are independent, with the notation . Two random bipartite networks are of particular interest:

  • The Bernoulli network: Here we assume that the random variables are independent Bernoulli random variables with a fixed parameter .

  • The complete network: In this case for all and , that is, all agents are linked to all objects and vice versa.

Multivariate hitting probabilities have been considered before with references given in the Introduction. This article focuses on the case that the sum of a non-empty selected subset of all components hits the sum of the barriers, and the case that all components in hit their barriers (an and-condition), that is

(2.5)

for such that . The case that at least one component hits its barrier (an or-condition) can be solved by an inclusion-exclusion argument and the and-condition. Obviously, this is only feasible for small networks, for bivariate compound Poisson models see [4, Eq. (7)].
If we simply denote while for for we write for . Hence while for any

Throughout we shall use the fact that the network does not change over time and is independent of the compound Poisson process.

3 The hitting probability of the aggregated risk process

As it will turn out, in the network the random variable, henceforth called the (network) Pollaczek-Khintchine parameter,

(3.1)

is the random equivalent of in the classical Pollaczeck-Khinchine formula (1.1) plays a crucial role in determining the hitting probability of sums of agents.

Remark 3.1.

If the weighted adjacency matrix is such that , then and . Hence the indicator in (3.1) is not mathematically necessary. Often we will keep this indicator for transparency; Example 3.5 illustrates a case where the indicator features prominently.

Theorem 3.2.

[Network Pollaczek-Khintchine formula for component sums]
For any the joint hitting probability

for a given level such that has representation

(3.2)

where is defined in (3.1) and

(3.3)
Proof.

By definition of the process we have

such that

For any realisation of the network with the process is a compound Poisson process with intensity, jump distribution and drift given by

Hence, whenever

for any fixed realisation of it holds that

by the classical Pollaczek-Khintchine formula (1.1). For the hitting probability is obviously . The result now follows by conditioning on the realisations of . ∎

Taking the point of view of a single agent we obtain the following as a direct consequence of Theorem 3.2. We detail this case for reference later.

Example 3.3.

[Network Pollaczek-Khintchine formula for one network component]
The hitting probability of a given level of for is given by

(3.4)

where

and

(3.5)

Comparing the classical Pollaczek-Khintchine formula (1.1) with the network versions in Theorem 3.2 and Example 3.3 above, in the network the role of in (1.1) is taken up by the random Pollaczek-Khintchine parameter from (3.1) with representation

(3.6)

In the following we collect some general observations on .

Remarks 3.4.
  1. Given it holds

    Thus, if all objects have a Pollaczek-Khintchine parameters , then . Nevertheless can be achieved even if some Pollaczek-Khintchine parameters exceed 1, as long as the others balance this contribution.

  2. Figures 2 and 3 illustrate the balancing effect in the setting of a homogeneous Bernoulli model, i.e. a Bernoulli network with edge probability and homogeneous weights as defined in Example 2.1. Even when half of the ’s are larger than 1, the mean of the Pollaczek-Khintchine parameter can still be considerably smaller than 1. This balancing effect increases with the size of the considered subset of agents as well as with the number of available objects.
    Also note that - as will be seen in Section 4.2 - for , independent of , behaves as , which, under the given set of parameters in Figures 2 and 3, has mean and standard deviation .
    As the distribution of is tedious to compute for large networks Figures 2, 3, and 4 below are based on Monte Carlo simulations. The conditioning on has been realized via the laws of total expectation and total variance.

  3. Eq. (3.6) shows that the Pollaczek-Khintchine parameter depends on the weights only through the ratios of sums of weights. Eq. (3.6) implies further that

    This bound is an equality when all agents are connected only to one single object. Otherwise the bound may be quite crude. Using the Markov inequality this bound can be used to bound for any :

Figure 2: (left) and (right) for as functions of in a homogeneous Bernoulli model with . The balancing effect of the network increases with the size of the considered subset of agents. The chosen parameters are for and .
Figure 3: as function of in a homogeneous Bernoulli model for varying numbers of agents (left) and varying numbers of objects (right). A higher number of objects results in an increased balancing effect of the network, because, for any given value , on average the agent connects to more objects. Nevertheless due to the homogeneity of the weights, the number of agents has no effect on the Pollaczek-Khintchine parameter. The chosen parameters are on the left: , for and ; right: , , and , , and , .
Example 3.5.

[Equal Pollaczek-Khintchine parameters]
If all are equal, we obtain directly from (3.1) that for any set

and hence for any measurable function on

(3.7)

In particular, if and only if . Comparing this condition to the condition in the non-network case, we see that the presence of the network allows for . The network thus balances the hitting probabilities for single components in the sense that is possible while still ensuring that .  

3.1 A Lundberg bound for

As in the classical one-dimensional setting, we expect exponential decay of the hitting probability of sums of agents also in the network setting when the jump distributions are light-tailed. To establish this, we start with a lemma which gives a sufficient condition for the existence of an adjustment coefficient for sums of components of the process as defined in (2.3).

Lemma 3.6.

Assume that for all the Pollaczek-Khintchine parameter and an adjustment coefficient exists, so that,

(3.8)

For every realisation of the network and , let where denotes the ’th component of the process with the realised network . Then for any such that , the process admits an adjustment coefficient satisfying

(3.9)

where

In particular,

(3.10)

Note that the condition in Lemma 3.6 just means that the degree of in the realisation is positive.

Proof of Lemma 3.6.

Recall from classical risk theory that has an adjustment coefficient satisfying (3.9) if and only if there exists such that

where the cumulant generating function is defined for all such that . By independence,

Since all contributing functions are convex and, by assumption (3.8), have a negative derivative in , the same holds true for . Hence there exists such that .
Finally, observe that for all implies , while for all implies that . ∎

We continue with a Lundberg inequality for the hitting probability of sums of components, generalizing (1.2) to the network setting. A similar result for hitting probabilities of sums of components of a multivariate risk process but without network structure is derived in [3, Ch. XIII, Proposition 9.3]. In order to find an adjustment coefficient which is independent of the specific realisation of the network, let be a constant such that for all ,

(3.11)

For example in a homogeneous system with we can take .

Theorem 3.7.

[Network bound for component sums]
Assume that for all the Pollaczek-Khintchine parameter and an adjustment coefficient satisfying (3.8) exists. Let be a subset of agents. Then

where and the are the bounds of the weights as given in (3.11).

Proof.

First note that the condition , entails . Thus by conditioning on the network realisation as in Theorem 3.2

By Lemma 3.6, for , has an adjustment coefficient and we know from the classical Lundberg estimate (1.2) that, assuming , for the network realisation ,

Clearly for all realisations such that

By (3.11) it follows from (3.10) that for all network realisations. Thus and we obtain the claimed bound. ∎

Note that the bound in Theorem 3.7 is optimal only in the case that the agents in are only connected to the objects with the heaviest tails in the jump distribution. For a given network structure one may obtain a Lundberg bound by determining the adjustment coefficient for the induced mixed jump distribution as long as the small jumps dominate, even including the possible case that some of the jump sizes are heavy tailed in the sense that (3.8) does not hold for all objects .

3.2 The exponential system

Assuming exponential jump sizes for all objects and weighted adjacency matrix (2.4), the joint hitting probability can be evaluated explicitly.

Theorem 3.8.

[Hitting probability for component sums]
Fix a subset of agents and assume an exponential system as defined in Example 2.2. Then the hitting probability of the sum of all agents in is given by

(3.12)

where

(3.13)
Proof.

We calculate the integrated tail distribution as in (3.3),

which is an exponential distribution function with mean . Hence is an Erlang distribution function with density

Moreover, due to the assumptions on the network, in (3.1) equals (3.13). From (3.2) we obtain for such that ,

(3.14)

Now we calculate that

and

Using this expression in (3.14) gives the assertion. ∎

Example 3.9.

[Hitting probability for a single agent]
Assume that the conditions of Theorem 3.8 hold, and let be a single agent. Then the hitting probability of agent for is given by

(3.15)

where

Remarks 3.10.
  1. Note that the argument in the expectation in (3.15) coincides with (1.3) with random .

  2. Note that (3.12) can be abbreviated as

    (3.16)

    with the function given as