Continuity and monotonicity of solutions to a greedy maximization problem

Motivated by an application to resource sharing network modelling, we consider a problem of greedy maximization (i.e., maximization of the consecutive minima) of a vector in $R^n$, with the admissible set indexed by the time parameter. The structure of the constraints depends on the underlying network topology. We investigate continuity and monotonicity of the resulting maximizers with respect to time. Our results have important consequences for fluid models of the corresponding networks which are optimal, in the appropriate sense, with respect to handling real-time transmission requests.


Introduction
Starting from seminal papers of Rybko and Stolyar [14], Dai [5], Bramson [2,3,4] and other authors, fluid models have become a standard tool in investigating long-time behaviour of complicated queueing systems. Such models are useful for establishing stability and obtaining hydrodynamic or diffusion limits for multiclass queueing networks and resource sharing networks with various service protocols. Using a similar methodology in the case of real-time Earliest Deadline First (EDF) networks with resource sharing is hindered by the lack of suitable fluid model equations. To overcome this difficulty, Kruk [13] suggested the definition of fluid models for these systems by means of an optimality property, called local edge minimality, which is known to characterize the EDF discipline in stochastic resource sharing networks. It turns out that the success of this approach depends on establishing suitable local properties of a vector-valued mapping F : [0, ∞) → R n , resulting from a greedy maximization (i.e., maximization of the consecutive minima) of a vector in R n over the admissible set A t , depending on the underlying network topology and indexed by the time parameter. A convenient way to describe the value F (t) for a given t ≥ 0 is to define it as the maximal element of A t with respect to a suitable "min-sensitive" partial ordering. Roughtly speaking, the function F , when well behaved, determines the so-called frontiers (i.e., the left endpoints of the supports) of the states in the corresponding locally edge minimal fluid model. The idea of using frontiers for the asymptotics of EDF systems dates back to the paper of Doytchinov et al. [7] on a G/G/1 queue, and it has been used several times since then. However, both the application of this idea to resource sharing networks and our approach to determining the frontiers by finding maximal elements of partially ordered sets appear to be new.
In this paper, for any t ≥ 0, we construct the value of F (t) as a solution of a nested sequence of max-min problems in a t-dependent admissible set. While none of these max-min problems is hard to solve, their number and forms vary with t in a complex, discontinuous way, making the analysis of the resulting mapping F on [0, ∞) rather involved. We investigate key properties of F , namely, its continuity and monotonicity. Our main results are described in more detail in Section 2.4, to follow, after the introduction of indispensable notation. These results are used in our forthcoming paper to establish fundamental properties of locally edge minimal fluid models, like their existence, uniqueness and stability.
We hope that the theory developed in this paper will be useful not only in the asymptotic analysis of EDF-like disciplines, but also in the case of other "greedy" scheduling policies for resource sharing networks, for example Longest Queue First [6] or Shortest Remaining Processing Time [15].

Notation
For sets A, B, we write A B if A is a proper subset of B. For a finite set A, let |A| denote the cardinality of A. Let N denote the set of positive integers and let R denote the set of real numbers. For a, b ∈ R, we write a ∨ b (a ∧ b) for the maximum (minimum) of a and b, a + for a ∨ 0. Vector inequalities are to be interpreted componentwise, i.e., for a, b ∈ R n , a = (a 1 , .., a n ), b = (b 1 , ..., b n ), a ≤ b if and only if a i ≤ b i for all i = 1, ..., n. For a = (a 1 , .., a n ) ∈ R n , we write min a for min i=1,...,n a i and Argmin a for {i ∈ {1, ..., n} : a i = min a}. For a = (a 1 , .., a n ) ∈ R n and a set A {1, ..., n} with {1, ..., n} \ A = {i 1 , ..., i k }, where k = n − |A| and i 1 < i 2 < ... < i k , we identify (a i ) i / ∈A with (a i 1 , ...a i k ) ∈ R k . By convention, a sum of the form m i=n ( m i=n ) with n > m or, more generally, a sum of numbers (resp., sets) over the empty set of indices equals zero (resp., Ø). For a set A ⊆ R, let A denote the closure of A.
2 The mapping F 2.1 "Min-sensitive" partial ordering on R n We define the relation " " inductively on R n as follows.
Definition 2.1 For a, b ∈ R, we write a b iff a ≤ b. If n ≥ 2, for a, b ∈ R n , a = (a 1 , ..a n ), b = (b 1 , ..., b n ), we write a b if one of the following four cases holds. The proof of the following lemmma is elementary and it is left to the reader.

Lemma 2.3
The relation " " is a partial ordering on R n . Remark 2.4 For a, b ∈ R n , the inequality a ≤ b implies a b. In dimensions greater than one the converse is in general, false, for example, (0, 2) (1, 1).

The mapping definition
Let I, J ∈ N and let I = {1, ..., I}, J = {1, ..., J}. For i ∈ I, let h i : R → R be a continuous, nonnegative, nondecreasing function with lim x→∞ h i (x) = ∞ and In other words, each h i is the cumulative distribution function of an atomless, σ-finite measure in R, with finite, nonpositive infimum of its support. Let G j , j ∈ J, be a family of distinct, nonempty subsets of I (not necessarily pairwise disjoint) such that I = j∈J G j .
Definition 2.5 For t ≥ 0, we denote by F (t) = (F i (t)) i∈I the maximal element of the set with respect to the relation " ".
Somewhat informally, F (t) may be thought of as the result of "greedy" maximization of a vector a ∈ R I , subject to the constraints defining the set A t , in the following sense. We first maximize min a over a ∈ A t , then we maximize the "next minimum" min{a i , i / ∈ Argmin a} over the set of maximizers of the previous problem, and we continue in this way until all the entries of the maximizer a * = F (t) are determined. In Section 3.1 we formalize and describe in detail this nested max-min procedure which implies, in particular, the existence and uniqueness of the maximizer F (t).
Remark 2.6 A seemingly more general version of Definition 2.5, in which for a fixed (possibly positive) t 0 ∈ R and t ≥ t 0 , the set A t is replaced by and the half-line (−∞, 0] in (2.1) is replaced by (−∞, t 0 ], may be easily reduced to the case considered in Definition 2.5 by the change of variables

Motivation: fluid models for resource sharing networks
The need for investigating the properties of the mapping F defined above arises in the theory of fluid models for real-time networks with resource sharing. Below, we briefly (and somewhat informally) describe this connection. The reader may consult [13] for more details and references. Consider a network with a finite number of resources (nodes), labelled by j = 1, ..., J, and a finite set of routes, labelled by i = 1, ..., I. Let I = {1, ..., I}, J = {1, ..., J}. For j ∈ J, let G j ⊆ I be the set of routes using the resource j. For convenience, we assume that all the resources have a unit service rate. By a flow on route i we mean a continuous transmission of a file through the resources used by this route. We assume that a flow takes simultaneous possession of all the resources on its route during the transmission. Each flow in the network has a deadline for transmission completion. Networks of this type may be used to model, e.g., voice and video transmission, manufacturing systems with order due dates or emergency health care services. In what follows, by the lead time of a flow we mean the difference between its deadline and the current time.
As in [12,13], the time evolution of such a system may be described by the process X(t, s) = (Z(t, s), D(t, s), T (t, s), Y (t, s)), t ≥ 0, s ∈ R, where the component processes Z, D, T , Y are defined as follows. For t ≥ 0 and s ∈ R, Z(t, s) = (Z i (t, s)) i∈I , where Z i (t, s) is the number of flows on route i with lead times at time t less than or equal to s which are still present in the system at that time. Similarly, the vectors D(t, s) = (D i (t, s)) i∈I , T (t, s) = (T i (t, s)) i∈I denote the number of departures (i.e., transmission completions) and the cumulative transmission time by time t corresponding to each route i of flows with lead times at time t less than or equal to s. Let Y i (t, s) = t − T i (t, s), i ∈ I, denote the cumulative idleness by time t with regard to transmission of flows on route i with lead times at time t less than or equal to s and let Y (t, s) = (Y i (t, s)) i∈I . The process X satisfies the following network equations, valid fort ≥ t ≥ 0, s ∈ R: where E(t, s) = (E i (t, s)) i∈I is the corresponding external arrival process and S i (t ′ , t, s) denotes the number of transmission completions of flows on route i ∈ I having lead times at time t less than or equal to s, by the time the system has spent t ′ units of time transmitting these flows. Fluid models are deterministic, continuous analogs of resource sharing networks, in which individual flows are replaced by a divisible commodity (fluid), moving along I routes with J resources (nodes). They usually arise from formal functional law of large numbers approximations of the corresponding stochastic flow level models. The analogs of the network equations (2.2)-(2.4) are the following fluid model equations, valid fort ≥ t ≥ 0, s ∈ R: where m i is the mean transmission time of a flow on route i and α = (α i ) i∈I is the vector of flow arrival rates. A system X(t, s) = (Z(t, s), D(t, s), T (t, s), Y (t, s)), t ≥ 0, s ∈ R, (2.8) with continuous, nonnegative components, satisfying the equations (2.5)-(2.7), together with some natural monotonicity assumptions, is called a fluid model for the resource sharing network under consideration.
To proceed further, we will introduce a class of fluid models which is, in some sense, optimal with respect to handling real-time transmission requests. To this end, we define a partial ordering "≪" on the space of real functions on R, which is extremely sensitive to the behaviour of the functions under comparison for small arguments.
Definition 2.7 ([13], Definition 5) Let f, g : R → R be such that for some a ∈ R we have f ≡ g on (−∞, a] and let c = sup{a ∈ R : f (x) = g(x) ∀x ≤ a}. We write f ≪ g if either c = ∞ (i.e., f ≡ g on R), or c < ∞ and there exists ǫ > 0 such that f ≤ g on [c, c + ǫ]. Definition 2.8 (see [13], Definition 11) A fluid model X of the form (2.8) for a resource sharing network with i∈I Z i (0, ·) ≡ 0 on (−∞, c] for some c ∈ R is called locally edge minimal at a time t 0 ≥ 0 if there exists h > 0 such that for any fluid model X ′ with the same α, m i , G i and the ) for every t ∈ (t 0 , t 0 + h). The fluid model X is called locally edge minimal, if it is locally edge minimal at every t 0 ≥ 0.
The intuition behind these notions is that a locally edge minimal fluid model tries to transmit as much "customer mass" corresponding to the earliest deadlines as possible, and hence its idleness with respect to such mass is as small as possible. Accordingly, such a model may be thought of as a "macroscopic" counterpart of a resource sharing network working under the Earliest-Deadline-First (EDF) protocol. Indeed, the EDF service discipline in such a network may be characterized by an analogous notion of local edge minimality, see [13], Definition 8 and Theorems 5-7.
In a forthcoming paper, we show that existence and local uniqueness of a fluid model for given data α, m i , G i , and initial state Z(0, ·), which is locally edge minimal at a time t 0 ≥ 0, is closely related to local monotonicity of the mapping F introduced in Section 2.2, with and suitably defined, not necessarily nonpositive, x * i , depending on t 0 (see Remark 2.6). It turns out that for large t 0 , the points x * i are also large, so under a natural assumption that the supports of Z i (0, ·) are bounded above, the functions h i in (2.9) are linear on [x * i , ∞). Consequently, the linear case, investigated in Section 5.1 of this paper, is of considerable importance, because it determines the long-time behaviour of the corresponding locally edge minimal fluid model, for example, its stability or the form of its invariant manifold. In particular, one of remarkable implications of the formulae developed in Section 5.1 is stability of locally edge minimal fluid models, regardless of the underlying resource sharing network topology. Results along this line may be found in our forthcoming paper

Overwiev of the main results
In this paper, we investigate key properties of the mapping F : [0, ∞) → R I , in particular those which are relevant to the theory of locally edge minimal fluid models. In Section 3, we present a detailed construction of F (t) and we provide some illustrating examples. The main result of Section 4 is Theorem 4.2, stating that if each function h i is strictly increasing in [x * i , ∞), then the corresponding map F is continuous. This basic regularity result, together with the method of partions, introduced in Section 3.3, is useful in proving various refinements, like Lipschitz continuity of F for Lipschitz h i (Section 5.2) or an upgrade of the local monotonicity result from linear to C 1 functions h i (Section 5.3). The main contribution of Section 5 is the explicit evaluation of the mapping F near zero in the linear case, implying, in particular, its local monotonicity in a neighbourhood of zero for piecewise linear h i . As we have already mentioned, the latter fact is then generalized to (piecewise) C 1 functions. Finally, in Section 5.4, we show that, somewhat surprisingly, the mapping F may fail to be globally monotone on [0, ∞), even if the corresponding functions h i are linear in [x * i , ∞).
3 The mapping construction algorithm

Construction
Fix t ≥ 0. We define the vector F (t) = (F i (t)) i∈I as follows. Let In what follows, we assume that f (1) < t. By continuity of h i , x = f (1) satisfies (3.1) and the set J (1) = j ∈ J : i∈G j h i (f (1) ) = t of active constraints is nonempty. (Indeed, if J (1) = Ø, then f = f (1) + ǫ also satisfies (3.1) for ǫ > 0 small enough, which contradicts the definition of f (1) .) Let I (1) = j∈J (1) G j and If I (1) = I, this completely determines the vector F (t). In this case, let k max = 1. Otherwise, let (2) )} for i ∈ I (2) . If I (1) ∪ I (2) = I, the definition of the vector F (t) is complete and we take k max = 2, otherwise we let ), and we continue our construction as follows.
Suppose that for some k ≥ 2, we have defined numbers and nonempty, disjoint subsets I (1) , ...,I (k) of I with k l=1 I (l) = I such that for l = 1, .., k, and K (l) = J \ l p=1 (J (p) ∪ N (p) ). Note that K (l) = Ø by (3.4) and G j \ l p=1 I (p) = Ø for j ∈ K (l) , l = 1, ..., k, by (3.5), so that the second sum in (3.6) is taken over a nonempty set of indices. (Such numbers and sets were defined in the last paragraph for k = 2.) Let f (k+1) = f (k+1) (t) be the supremum of x ≤ t satisfying the constraints , so the definition of the vector F (t) is complete. In this case, we put k max = k + 1. If f (k+1) < t, then x = f (k+1) satisfies (3.8) and the set J (k+1) of active constraints in (3.8) (i.e., these j ∈ K (k) , for which equality holds in (3.8) with x = f (k+1) ) is nonempty. In this case, let . This ends the k + 1-th step of our construction. If k+1 l=1 I (l) = I, the definition of the vector F (t) is complete. In this case, put k max = k + 1. Otherwise, we make another (i.e., the k + 2-th) step of our algorithm, taking k + 1 instead of k and proceeding as above.
When the construction terminates after k max steps, we have defined the vector F (t).
Remark 3.1 The index k max and the sets I (k) , J (k) , N (k) , K (k) defined above depend on the time t. In what follows, when we want to stress this dependence, we write k max (t), In general, some of the sets N (k) , k = 1, ..., k max , may be nonempty, see Example 3.5, to follow. If j ∈ N (k) for some k, then This strict inequality may be interpreted as an indication of "unavoidable bottleneck idleness" in the corresponding locally edge minimal fluid network -transferring higher priority fluids by other resources does not allow j to use its full capacity on the time interval [0, t]. This phenomenon is well known in the theory of resource sharing networks and it was discussed in detail, e.g., by Gurvich and Van Mieghem [8,9]. A mild sufficient condition for all the sets N (k) to be empty is that for each j ∈ J, G j \ j ′ =j G j ′ = Ø. This corresponds to the so-called local traffic condition for the underlying network topology, under which every resource has at least one route using only that resource, see [10,11]. The latter requirement is satisfied, for example, by linear networks, for which I = J + 1 and There are, however, some important systems that do not satisfy the local traffic assumption, for example ring networks, used as counterexamples for stability of the LQF protocol [1,6].

Examples
In this subsection, we provide two examples illustrating the construction of the mapping F : [0, ∞) → R I defined in Subsection 3.1. The first one has relatively simple structure, yielding time-independent k max , J (k) , I (k) , N (k) in (0, ∞) and linear function F . The second one, in which k max , J (k) , I (k) vary in time and F is nonlinear, indicates some of the difficulties encountered in more general situations.
In what follows, we assume that t > 0. Then In the remainder of this example, the time argument in J (k) (t), be given by (3.2). For j belonging to the set Moreover, J (2) is the set of j ∈ K (1) attaining the minimum in (3.11) and (3.11) is attained at every j ∈ K (1) ), then the construction of F is complete. Otherwise we proceed similarly, until we get f (3) (t),...,f (kmax) (t), and hence all F i (t), i ∈ I, in the form of linear functions of t, with slopes depending (in an increasingly complicated way) on the sets G j , describing the topology of the corresponding network. Observe that in this example some sets N (k) may, in general, be nonempty. In general, the main difficulty in analyzing the properties of the mapping F , already indicated in Remark 3.1, is the time-dependence of the index k max and the sets I (k) , J (k) , k = 1, ..., k max . The following example, corresponding to a simple linear network topology, illustrates this point. Note that, by Remark 3.4, in this case the sets N (k) are empty. (3.14) It is easy to see that the mapping F given by (3.12)-(3.14) is continuous and nondecreasing on [0, ∞). This example also indicates that there is, in general, no hope for obtaining any global results about the auxiliary functions f (p) , p > 1. Here, f (2) is linear and strictly increasing in (1, 7/2) and (7/2, ∞), but it does not even exist in [0, 1] ∪ {7/2}.

Partitions and inverses
Let J be the set of "ordered partitions" of the set J, i.e., finite sequences of subsets of J in the form ( where the sets N 1 , ..., N k are defined recursively as We label the nonempty sets of the form T D , D ∈ J , as T 1 , ...T d . Clearly, For a function g : R → [0, ∞) with lim x→∞ g(x) = ∞, let g −1 denote its (generalized) right-continuous inverse (see, e.g., [16], Section 13.6), If the function g is nondecreasing and g(x 0 ) = 0 for some x 0 ∈ R, then For t ∈ T D , we can write f (p) (t), p = 1, ..., k max (t), (and hence F (t)) in closed form, using the inverse functions introduced above. To see this, let t ∈ T D and note that k = k max (t). Choose j 1 , ..., j k ∈ J so that j p ∈ J p = J (p) (t) for each j = 1, ..., k. Then and for p = 2, ..., k, we have the recursive formulae where

Continuity
In general, the function F may have jumps, as the following one-dimensional example indicates.
Clearly, the discontinuity of F 1 = f (1) at t = 2 in this example is caused by a "flat spot", i.e., the interval [1, 2] on which h 1 takes the constant value 2, resulting in the jump of h −1 1 (see (3.16)). In our queueing application, this corresponds to the lack of "customer mass" with deadlines in the interval [1,2] in the system, causing the frontier to jump over this empty interval. The following theorem assures that in the absence of such "flat spots", the function F is actually continuous. Theorem 4.2 Assume that for every i ∈ I, the function h i is strictly increasing in [x * i , ∞). Then the mapping F is continuous in [0, ∞). In the proof of this result, we will use the following elementary lemma. For the sake of completeness, we provide its justification. Lemma 4.3 Let (X, d) be a metric space. Let y ∈ X and let {x n } be a sequence of elements of X such that every subsequence {x n k } of {x n } contains a further subsequence {x n k l } converging to y. Then lim n→∞ x n = y.
Proof. Suppose that the sequence {x n } does not converge to y. This means that there exist ǫ > 0 and a subsequence {x n k } of {x n } such that d(x n k , y) ≥ ǫ for all k. However, we have assumed the existence of a subsequence {x n k l } of {x n k } such that lim l→∞ d(x n k l , y) = 0, so we have a contradiction. ✷ The proof of Theorem 4.2 is long and somewhat involved, so we only sketch it here, moving most of the technical details to the appendix. It is convenient to introduce additional notation Clearly, the sets D (p) = D (p) (t) depend on the time t, see Remark 3.1.
Sketch of the proof of Theorem 4.2. Let t n > 0 be such that t n → t 0 as n → ∞. We may assume that t n ≤ t 0 + 1 for all n. Our aim is to show that F (t n ) → F (t 0 ), i.e., for every i ∈ I, we have Without loss of generality (passing to a subsequence if necessary) we may assume that for every m, n ≥ 1, we have k max (t m ) = k max (t n ) and J (p) (t m ) = J (p) (t n ) (hence I (p) (t m ) = I (p) (t n ), N (p) (t m ) = N (p) (t n )) for p = 1, ..., k max (t m ).
Consequently, in what follows, we will simply write k max , instead of k max (t n ), n ≥ 1, J (p) instead of J (p) (t n ), n ≥ 1, e.t.c.. By definition, for n ≥ 1 and p = 1, ..., k max , min i∈I x * i ≤ f (p) (t n ) ≤ t n ≤ t 0 + 1. Hence, by Lemma 4.3, without loss of generality (passing to a subsequence if necessary) we may assume that the sequences {f (p) (t n )}, p = 1, ..., k max , converge. Let First suppose that t 0 = 0. We have J (1) (0) = J, I (1) (0) = I, k max (0) = 1 and F i (0) = x * i for all i ∈ I. Let i ∈ I. Then i ∈ I (k) for some k ∈ {1, ..., k max }. By the definition of f (k) , 0 ≤ h i (f (k) (t n )) ≤ t n , and hence, as t n ↓ 0, by (2.1), so F i is continuous at 0. Hence, in what follows, we will assume that t 0 > 0. We will first consider the case in which f (1) (t n ) = t n for all n ≥ 1. Then f (1) (t 0 ) = t 0 and thus, for each i ∈ I, . Therefore, we may additionally assume f (1) (t n ) < t n , n ≥ 0. (4.6) We prove (4.2) inductively for i ∈ I (l) (t 0 ), l = 1, ..., k max (t 0 ). In the l-th inductive step, we define It is easy to check that where B We also argue that ifp l < p l , then for i ∈ B (l) Finally, the equations (4.9), (4.11)-(4.12) imply (4.2) for every i ∈ I (l) (t 0 ). The details of the above inductive argument may be found in the Appendix.

Monotonicity
In this section, we investigate monotonicity of the mapping F . It turns out that, in general, F fails to be globally nondecreasing, even if the functions h i are piecewise linear, see Section 5.4. However, under suitable assumptions on h i , monotonicity of F in some neighbourhood of 0 may be established. More precisely, our goal is to find T > 0 such that for every 0 ≤ t <t < T and i ∈ I, we have This is done in Section 5.1 for piecewise linear functions h i , i ∈ I.
Remark 5.1 Since the above argument is local in time, it actually requires only that for each i ∈ I, (5.2) holds in some neighborhood of x * i , i.e., for all x < X * i , where X * i > x * i are given constants. (Without loss of generality we may further assume that We only have to restrict t in each step of our construction to the interval [0, In this case, (5.1) holds for all i ∈ I and 0 ≤ t <t < t ′ kmax .

Reduction lemmas and Lipschitz continuity
In this subsection we consider the case of general h i , assuming only that each h i is strictly increasing in [x * i , ∞). Fix T > 0 and recall the sets T 1 , ...T d from Subsection 3.3. The following lemma reduces the problem of establishing monotonicity of F to showing its monotonicity under the additional assumption that k max and the sets J (p) , I (p) , p = 1, ..., k max , are constant in t.
Lemma 5.2 Assume that for k = 1, ..., d, the mapping F is nondecreasing on T k . Then F is nondecreasing on [0, T ].
Proof. First note that F is nondecreasing on T k for each k = 1, ..., d. Indeed, let t,t be such that t <t and t,t ∈ T k for some k. Then for each n ∈ N, there exist t n ,t n ∈ T k such that t n <t n and t n → t,t n →t as n → ∞. By assumption, F (t n ) ≤ F (t n ) for each n. Letting n → ∞, by Theorem 4.2 we get F (t) ≤ F (t), so F is indeed monotone on T k .
Let 0 ≤ t <t ≤ T and let k 0 be such that t ∈ T k 0 . Let t 1 = sup{s ≤ t : s ∈ T k 0 }. Then t 1 ∈ T k 0 , so F (t) ≤ F (t 1 ). If t 1 =t, we have (5.1) and the proof is complete. Assume that t 1 <t. In this case, (t 1 ,t] ∩ T k 0 = Ø by the definition of t 1 . However, there exist k 1 = k 0 and a sequence s n ∈ T k 1 such that s n ↓ t 1 . Let t 2 = sup{s ≤t : s ∈ T k 1 }. Then t 1 , t 2 ∈ T k 1 , so F (t 1 ) ≤ F (t 2 ), and hence F (t) ≤ F (t 2 ). If t 2 =t, the proof is complete, otherwise (t 2 ,t] ∩ (T k 0 ∪ T k 1 ) = Ø. Let k 2 / ∈ {k 0 , k 1 } and a sequence s n ∈ T k 2 be such that s n ↓ t 2 . Put t 3 = sup{s ≤t : s ∈ T k 2 }. As above, we have F (t 2 ) ≤ F (t 3 ), and hence F (t) ≤ F (t 3 ). After a finite number l ≤ d of such steps we get F (t) ≤ F (t l ) and t l =t, so (5.1) holds. ✷ A slight modification of the above argument yields Lemma 5.3 Assume that for k = 1, ..., d, the mapping F is Lipschitz con- This lemma, together with (3.16)-(3.18), may be used to provide a simple proof of the following result.
Theorem 5.4 Assume that there exist 0 < c < C < ∞ such that

Local monotonicity in the C 1 case
In this subsection, we assume that for each i ∈ I, h i ∈ C 1 ([x * i , ∞)) and where h ′ i (x * i ) denotes the right derivative of h i at x * i . Again, with no loss of generality we may assume (5.3). Define m * , n 0 , ..., n m * , y * 1 , ..., y * m * as in Section 5.1. Because our concern here is local monotonicity of F , without loss of generality we may assume that h ′ i (x) > 0 for all i ∈ I and x ≥ x * i (compare Remark 5.1).
The main idea of our analysis for this case is to consider it as a small perturbation of the linear problem considered in Section 5.1, with ρ i given by (5.16). In particular, let t kmax be as in in Section 5.1 and let k L max be the constant k max (t), 0 < t < t kmax , defined there. Furthermore for k = 1, ..., k L max , let a (k) be as in Section 5.1 and let J (k) L , denote the sets J (k) (t), I (k) (t), N (k) (t), K (k) (t), t ∈ (0, t kmax ), and the function f (k) for the above-mentioned linear problem, respectively.
Fix 0 < T ≤ t kmax /2 and the partition D = (J 1 , ..., J k ) ∈ J such that there exists a sequence t n ↓ 0 such that t n ∈ T D for all n. Since f (1) (t) < ... < f (k−1) (t) < t for t ∈ T D , using (3.16)-(3.17) and proceeding by induction, it is easy to see that f (1) ,..., f (k−1) are the truncations of C 1 functions to T D , and f (k) is the truncation to T D of a function in the form g(t) ∧ t, where g is C 1 . Therefore, in order to prove that the functions f (p) , p = 1, ..., k = k max , (and hence, by (3.10), F i , i ∈ I) are nondecreasing in an intersection of T D and a neighborhood of zero, it suffices to check that where f (p) (0) := f (p) (0+) = lim n→∞ f (p) (t n ). Consequently, by Lemma 5.2, if we verify (5.17), then the proof of monotonicity of the mapping F in a neighborhood of zero is complete. In what follows, we consider only t ∈ T D .

Lack of global monotonicity
It is not hard to prove that, without any additional assumptions on the functions h i , (5.1) holds for every 0 ≤ t <t and i ∈ I (1) (t). In spite of this, the mapping F is, in general, not monotone on [0, ∞), even if h i , i ∈ I, are given by (5.2), as the following example shows.

Further research directions
The results obtained in this paper may be regarded as introductory in nature and there are several important issues regarding our mapping F that remain to be addressed. First, one would like to relax the assumptions on regularity of h i necessary for local monotonicity of F . Example 5.5 shows that "kinks" of h i may create problems in this regard, so it is not immediately clear that the monotonicity result of Section 5.3 may be carried over even to the Lipschitz case. A remedy for this problem may be creating a "differential" version of the algorithm from Section 3.1, determining the derivatives of f (p) , rather than their values, for Lipschitz h i , in a way similar to our analysis for the linear case. This, however, in the absence of C 1 regularity of h i , yields an ODE system with discontinuous right-hand side, so even establishing existence of solutions to such a system may be challenging.
Another direction that appears to be important for applications is to skip the assumption of strict monotonicity of h i . As Example 4.1 indicates, this results in jumps of the corresponding process F . However, from the point of view of the queueing application described in Section 2.3, with the functions h i given by (2.9), this is not necessarily a problem, because some F i may "jump over the flat spots", containing no mass of the corresponding initial distributions Z i (0, ·), without causing discontinuity of the resulting locally edge minimal fluid model. Similarly, it may be useful to investigate functions h i with upward jumps, corrresponding to distributions with atoms. This would open an avenue to using techniques similar to those developed in our forthcoming paper, but for pre-limit stochastic networks, rather than for the corresponding fluid limits.
Example 5.5 shows that there is no hope for global monotonicity of F in the general case. However, it is plausible that for some simple network topologies (e.g., linear or tree networks), the mapping F is monotone on [0, ∞). This would greatly simplify the analysis of the corresponding fluid limits, and aid the investigation of the pre-limit stochastic networks.
Finally, it may be interesting to replace the relation " " and/or the set A t in the definition of F (t) by a different partial ordering and/or admissible set, and to investigate properties and possible applications of the resulting mappings.
7 Appendix: Inductive proof of Theorem 4.2 We continue the argument starting in Section 4. As we have already explained, we may assume that t 0 > 0 and that (4.3), (4.6) hold.