# Dominant poles and tail asymptotics in the critical Gaussian many-sources regime

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## Abstract

The dominant pole approximation (DPA) is a classical analytic method to obtain from a generating function asymptotic estimates for its underlying coefficients. We apply DPA to a discrete queue in a critical many-sources regime, in order to obtain tail asymptotics for the stationary queue length. As it turns out, this regime leads to a clustering of the poles of the generating function, which renders the classical DPA useless, since the dominant pole is not sufficiently dominant. To resolve this, we design a new DPA method, which might also find application in other areas of mathematics, like combinatorics, particularly when Gaussian scalings related to the central limit theorem are involved.

## Keywords

Heavy traffic Many sources Asymptotics Dominant pole approximation Saddle point method QED regime## Mathematics Subject Classification

60K25 80M35## 1 Introduction

Probability generating functions (PGFs) encode the distributions of discrete random variables. When PGFs are considered analytic objects, their singularities or poles contain crucial information about the underlying distributions. Asymptotic expressions for the tail distributions, related to large deviations events, can typically be obtained in terms of the so-called dominant singularities, or dominant poles. The dominant pole approximation (DPA) for the tail distribution is then derived from the partial fraction expansion of the PGF and maintaining from this expansion the dominant fraction related to the dominant pole. Dominant pole approximations have been applied in many branches of mathematics, including analytic combinatorics [7] and queueing theory [19]. We apply DPA to a discrete queue that has an explicit expression for the PGF of the stationary queue length. Additionally, this queue is considered in a many-sources regime, a heavy-traffic regime in which both the demand on and the capacity of the systems grow large, while their ratio approaches one. This many-sources regime combines high system utilization and short delays, due to economies of scale. The regime is similar in flavor to the QED (quality and efficiency driven) regime for many-server systems [8], although an important difference is that our discrete queue fed by many-sources falls in the class of single-server systems and therefore leads to a manageable closed form expression for the PGF of the stationary queue length *Q*. Denote this PGF by \(Q(z)=\mathbb {E}(z^Q)\).

*Q*(

*z*) is larger than one (in which case all moments of

*Q*exist). This radius of convergence is in fact determined by the dominant singularity \(Z_{0}\), the singularity in \(|z|>1\) closest to the origin. For PGFs, due to Pringsheim’s theorem [7], \(Z_{0}\) is always a positive real number larger than one. Then DPA leads to the approximation

*N*large) for the tail probabilities \(\mathbb {P}(Q> N)\). We shall now explain in more detail the many-sources regime, the discrete queue, and when combining both, the mathematical challenges that arise when applying DPA.

### 1.1 Many sources and a discrete queue

*A*, with mean \(\mu _A\) and variance \(\sigma ^2_A\). For systems facing large demand, one can set the capacity according to the rule \(s=\mu _A+\beta \sigma _A\), which consists of a minimally required part \(\mu _A\) and a variability hedge \(\beta \sigma _A\). Such a rule can lead to economies of scale, as we will now describe in terms of a setting in which the demand per period is generated by many sources. Consider a system serving

*n*independent sources and let

*X*denote the generic random variable that describes the demand per source per period, with mean \(\mu \) and variance \(\sigma ^2\). Denote the service capacity by \(s_n\), so that the system utilization is given by \(\rho _n=n\mu /s_n\), where the index

*n*expresses the dependence on the scale at which the system operates. The traditional capacity sizing rule would then be

*n*with associated loads \(\rho _n\) such that (also using that \(s=s_n\sim n\mu \))

*A*, with mean \(n\mu \) and variance \(n\sigma ^2\). The capacity per period \(s_n\) is fixed and integer valued. The scaling rule in (1.3) thus specifies how the mean and variance of the demand per period, and simultaneously \(s_n\), will all grow to infinity as functions of

*n*. Many-sources scaling became popular through the Anick–Mitra–Sondhi model [1], as one of the canonical models for modern telecommunications networks, in which a switch may have hundreds of different input flows. But apart from communication networks, the concept of many sources can apply to any service system in which demand can be regarded as coming from many different inputs (see, for example, [4, 6, 13, 17, 18] for specific applications).

### 1.2 How to adapt classical DPA?

*dominated*poles: all poles other than the dominant pole. Moreover, the dominant pole itself approaches 1 according to

*N*, the factor \(Z_{0}^{-N-1}\) goes to the degenerate value 1. The many-sources regime thus has a fascinating effect on the location of the poles that renders a standard DPA useless for multiple reasons. We shall therefore adapt the DPA in order to make it suitable to deal with the complications that arise in the many-sources regime, with the goal to again obtain an asymptotic expansion for the tail distribution. First observe that the term \(Z_{0}^{-N-1}\) in (1.1) becomes non-degenerate when we impose that \(N\sim K\sqrt{n\sigma ^2}\), with

*K*some positive constant, in which case

*N*and

*n*can be coupled, but due to (1.4), only couplings for which

*N*is proportional to \(\sqrt{n}\) lead to a non-degenerate limit for \(Z_{0}^{-N-1}\). Now let us turn to the other two remaining issues: the fact that \(c_0/(1-Z_{0})\) potentially explodes and that the dominated poles converge to the dominant pole.

*Q*(

*z*) of

*Q*along a circle \(|z|=1+\varepsilon <Z_0\). By Cauchy’s theorem, the tail probabilities can be written as a residue \(c_0/(1-Z_0)Z_0^{N+1}\) at \(z=Z_0\) and an integral along a contour shifted beyond \(Z_0\). Using the product expansion of

*Q*(

*z*), involving the zeros of a characteristic equation in \(|z|\le 1\), the quantity \(c_0/(1-Z_0)\) can be approximated in terms of an integral of the Pollaczek type that has been considered in [11]. In [11], a dedicated saddle point method has been developed to approximate Pollaczek-type integrals for the mass at zero, the mean, and the variance of

*Q*in the many-sources regime. This dedicated saddle point method can also be used to find a many-sources approximation for \(c_0/(1-Z_0)\). The remaining challenge is then to bound the contribution of the contour integral shifted beyond the dominant pole \(Z_0\). This depends on how far the integration path can be shifted, and this is determined by the positions of the first dominated poles. It turns out that, in the many-sources regime, the dominated poles have approximations of the type (1.4) as well. The integration path is then chosen, roughly, as the circle that passes halfway between \(Z_0\) and the first dominated pole. This, together with the dedicated saddle point method to approximate \(c_0/(1-Z_0)\), then provides a fully rigorous derivation of the asymptotic expression for \(\mathbb {P}(Q> N)\) that is of the form

*n*be well approximated by the tail distribution of the reflected Gaussian random walk. We return to this connection in Sect. 5.

Our approach thus relies on detailed knowledge about the distribution of all the poles of the PGF of *Q*, and in particular how this distribution scales with the asymptotic regime (1.2)–(1.3). As it turns out, in contrast with classical DPA, this many-sources regime means that all poles contribute to the asymptotic characterization of the tail behavior. Our saddle point method leads to an asymptotic expansion for the tail probabilities, of which the limiting form corresponds to the heavy-traffic limit, and pre-limit forms present refined approximations for pre-limit systems (\(n<\infty \)) in heavy traffic. Such refinements to heavy-traffic limits are commonly referred to as *corrected diffusion approximations* [2, 3, 14]. Compared with the studies that directly analyzed the Gaussian random walk [5, 9, 10], which is the scaling limit of our queue in the many-sources regime, we start from the pre-limit process description and establish an asymptotic result which is valuable for a queue with a finite yet large number of sources. Starting this asymptotic analysis from the actual pre-limit process description is mathematically more challenging than directly analyzing the process limit, but in return gives valuable insights into the manner and speed at which the system starts displaying its limiting behavior.

### 1.3 Outline of the paper

In Sect. 2 we describe the discrete queue in more detail and present some preliminary results for its stationary queue length distribution. In Sect. 3 we give an overview of the results and the contour integration representation for the tail distribution. In Sect. 4 we give a rigorous proof of the main result for the leading-order term using the dedicated saddle point method (Subsect. 4.1), and we bound the contour integral with integration path shifted beyond the dominant pole (Subsect. 4.2). In Sect. 5 we elaborate on the connection between the discrete queue and the Gaussian random walk, and we present an asymptotic series for \(\mathbb {P}(Q >N)\) comprising not only the dominant poles but also the dominated poles.

## 2 Model description and preliminaries

*A*. The system has a service capacity \(s\in \mathbb {N}\) per period, so that the recursion

*A*exist.

*n*sources, \(X_{1,k}+...+X_{n,k}\), where the \(X_{i,k}\) are, for all

*i*and

*k*, i.i.d. copies of a random variable

*X*, of which the PGF \(X(z)=\sum _{j=0}^{\infty }\mathbb {P}(X=j)z^j\) has radius of convergence \(r>1\), and

*Q*defined as having this stationary distribution. We let

It is a well-known consequence of Rouché’s theorem that under (2.3) \(z^s-A(z)\) has precisely *s* zeros in \(|z|\le 1\), one of them being \(z_0=1\). We proceed in this paper under the same assumptions as in [11]. Thus, we assume that \(|X(z)|<X(r_1)\), \(|z|=r_1\), \(z\ne r_1\), for any \(r_1\in (0,r)\); see [11], end of Sect. 2 for a discussion of this condition. Finally, we assume that the degree of *X*(*z*) is larger than *s* / *n*. Under these conditions, \(z_0\) is the only zero of \(z^s-A(z)\) on \(|z|=1\), and all others in \(|z|\le 1\), denoted by \(z_1,z_2,...,z_{s-1}\), lie in \(|z|<1\). Furthermore, there are at most countably many zeros \(Z_k\) of \(z^s-A(z)\) in \(1<|z|<r\), and there is precisely one, denoted by \(Z_0\), with minimum modulus. The zeros \(Z_k\) are indexed using integer *k*; they come in conjugate pairs and we let \(Z_k=Z_{-k}^*\) where \(Z_k\) is in the (closed) lower half plane for non-negative *k*. For our analysis, it is crucial that \(r>1\), and so certain heavy-tailed distributions for *X* (with \(r=1\)) are excluded.

*Q*(

*z*) to all

*z*, \(|z|<r\) and

*z*not a zero of \(z^s-A(z)\), where the right-hand side of (2.5) is analytic in \(|z|<Z_0\) and has a first-order pole at \(z=Z_0\). We have, for the tail probability (using that \(Q(1)=1\)), for \(N=0,1,...\)

*f*(

*z*). By contour integration, Cauchy’s theorem and \(Q(1)=1\), we then get for \(0<\varepsilon <Z_0-1\)

*R*is any number between \(Z_0\) and \(\min _{k\ne 0}|Z_k|\). When

*n*and

*s*are fixed, we have that the integral on the second line of (2.7) is \(O(R^{-N})\), and so there is the DPA

*R*and \(Z_0\) in (2.7) tend to 1, and thus both terms on the second line of (2.7) need special attention. The second term here is approximated, using the product form expansion in (2.5), in terms of a contour integral of the Pollaczek type to which the dedicated saddle point method of [11], Sect. 3, can be applied. The method has been developed in [11] from Pollaczek’s integral representation of the PGF of

*Q*,

*Q*in the generalized heavy-traffic regimes \(n/s=1-\gamma n^{-\alpha }\), \(\alpha >0\), and \(n\rightarrow \infty \). To this end, the contour integrals are deformed so as to pass through the saddle point \(z_\mathrm{sp}\in (1,Z_0)\) given as the zero

*z*of \(g'(z)\), where \(g(z)=-\ln z +\frac{1}{s}\ln A(z)={-\ln z+\frac{n}{s}\ln X(z)}\). A crucial ingredient here is the substitution \(z=z(v)\), \(-\frac{1}{2} \delta \le v \le \frac{1}{2} \delta \) for some \(\delta >0\), such that

*s*one has

*B*positive and bounded away from 0 and 1, and \(\eta \) positive and bounded away from 0.

## 3 Overview and results

*n*and the capacity \(s=s_n(\gamma )\):

### Lemma 3.1

Due to this clustering phenomenon, the main reasoning that underpins classical DPA cannot be carried over. Starting from the expression (2.8), we need to investigate what becomes of the term \(c_0/(1-Z_0)\), and moreover, the validity of the exponentially small phrase in (2.8) and the actual *N*-range both become delicate matters that need detailed information about the distribution of the zeros as in Lemma 3.1.

Let us first present a result that identifies the relevant *N*-range:

### Proposition 3.2

### Proof

*L*is bounded away from 0 and \(\infty \), and this gives the result. \(\square \)

### Lemma 3.3

*P*(

*z*), in Lemma 3.4 below we evaluate \(\ln P(Z)\) for \(|Z|\ge 1\) in terms of the contour integral

### Lemma 3.4

*I*(

*Z*), with saddle point \(z_\mathrm{sp}=1+\varepsilon \) of the function \(g(z)={-}\ln z+\frac{n}{s} \ln (X(Z))\), yields

### Proposition 3.5

*R*appropriately. To do this, we consider the product representation (2.5) of

*Q*(

*z*), and we want to choose

*R*such that \(|z^s-A(z)|\ge C |z|^s\), \(|z|=R\), for some \(C>0\) independent of

*s*. It will be shown in Sect. 4 that this is achieved by taking

*R*such that the curve \(|z^s|=|A(z)|\), on which \(Z_0\) and \(Z_{\pm 1}\)(\(=Z_{+1},Z_{-1}\)) lie, is crossed near a point

*z*(also referred to as \(Z_{\pm 1/2}\)) where \(z^s\) and

*A*(

*z*) have opposite sign. A further analysis, using again the dedicated saddle point method to bound the product \(\prod _{j=1}^{s-1}\) in (2.5), then yields that the integral in (3.16) decays as \(R^{-N}\). Finally, using the asymptotic information in (3.2)–(3.4) for \(Z_0\) and \(Z_{\pm 1}\), with \(Z_{\pm 1/2}\) lying midway between \(Z_0\) and \(Z_{\pm 1}\), the integral on the second line of (2.7) can be shown to have relative order \(\exp ({-}DN/\sqrt{s})\), for some \(D>0\) independent of

*s*, compared to the dominant pole term in (2.8).

*s*, \(N=1,2,\ldots \). The DPA \(c_0/(1-Z_{0})^{-1}Z_{0}^{-N-1}\) of \(\mathbb {P}(Q> N)\) thus has a relative error that decays exponentially fast.

*Q*, considered in the many-sources regime, is shown to be connected to the Gaussian random walk. This connection will imply that the front factor of the DPA in (3.17) satisfies

*L*bounded away from 0 and \(\infty \). The leading term in (3.19) agrees with (1.6) when we identify

*R*of the integration contour in (3.16) to \(R_M\) such that the poles \(Z_0, Z_{\pm 1},\ldots ,Z_{\pm M}\) are inside \(|z|=R_M\). This leads to

*L*bounded away from 0 and \(\infty \), we find from (3.22) and Proposition 3.2 that

## 4 DPA through contour integration

In this section we present the details of getting approximations of the tail probabilities using a contour integration approach as outlined in Sect. 3. In Subsect. 4.1 we concentrate on approximation of the front factor \(c_0/(1-Z_0)\) and the dominant pole \(Z_0\), and combine these to obtain an approximation of the leading-order term in (2.8). This gives Lemmas 3.3 and 3.4, and Proposition 3.5.

In Subsect. 4.2 we assess and bound the integral on the second line of (2.7) and thereby make precise what exponentially small in (2.8) means in the present setting.

### 4.1 Approximation of the leading-order term

#### 4.1.1 Proof of Lemma 3.3

#### 4.1.2 Proof of Lemma 3.4

#### 4.1.3 Proof of Proposition 3.5

*I*(

*Z*), where we take \(\varepsilon \) such that

*I*(

*Z*) comes from the

*z*’s in (3.12) close to \(z_\mathrm{sp}\).

### 4.2 Bounding the remaining integral

*Q*(

*z*) as represented by the right-hand side of (2.5) which is defined and analytic in

*z*, \(|z|<r\), \(z\ne Z_k\). We write for \(|z|<r\), \(z\ne Z_k\)

*s*. Hence \((s-\mu _A)/\prod _{j=1}^{s-1} (1-z_j)\) is bounded in

*s*. Next, for \(|z|\ge Z_0\), we have by Lemma 3.4

*I*(

*z*) given by (3.12) and admitting an estimate

*s*, such that

*z*is on a contour

*K*as in Fig. 1, consisting of a straight line segment

*s*, approximates the solution \(z=Z(t)\), for real

*t*small compared to

*s*, of the equation

*K*, we have from (4.31)

*s*. Observing that

*s*. Hence, by (4.36), we see that the relative error in (4.27) due to ignoring the integral on the right-hand side is of order \(\exp ({-}DN/\sqrt{s})\) with some \(D>0\), independent of

*s*.

*X*: there is a \(\delta >0\) and a \(\vartheta _1\in (0,\pi /2)\) such that for any \(R\in [1,1+\delta ]\) the function \(|X(R e^{i\vartheta })|\) is decreasing in \(|\vartheta |\in [0,\vartheta _1]\) while

*X*(

*z*) in the disk \(|z|<r\) (with \(r>1\)).

*K*in (4.31–4.32), we consider the quantity

*v*is of the form

*Z*(

*t*) is crossed by \(z=1+v\) near the points \(Z({\pm }\frac{1}{2})\), where \(z^s-A(z)\) equals \(2z^s\). Thus, we choose

*s*, see (4.45), the leading part of the right-hand side in (4.46) is independent of

*s*and describes, as a function of the real variable

*y*, a parabola in the complex plane with real part bounded from above by its real value at \(y=0\) and that passes the imaginary axis at the points \(\pm \pi i\). Therefore, this leading part has a positive distance to all points \(2\pi ik\), integer

*k*. Now take \(y_0\) such that

*K*(bold), the approximation \(\hat{Z}(t)\) of the outer curve, and the choice \(y_0=\eta _0\sqrt{s}\) for the case that \(\gamma =1\), \(\mu /\sigma ^2=2\), \(s=100\).

*v*as in (4.44), that

*c*is bounded away from 1 and \(|c|<1\). Now write

*s*is large enough, we have that \(R\in [1,1+\delta ]\) and \(0\le \vartheta _0\le \vartheta _1\), where \(\delta \) and \(\vartheta _1\) are as above in (4.42). We have

*K*with

*s*gets large.

## 5 Correction terms and asymptotic expansion

In this section, we give a series expansion for the leading term in (3.15) involving the Riemann zeta function. We also show how to find an asymptotic series for \(P(Q>N)\) as \(N\rightarrow \infty \) of which the term involving the dominant pole is the leading term. Before we do so, we first discuss how this leading term is related to the Gaussian random walk and a result of Chang and Peres [5].

### 5.1 Connection with Gaussian random walk

*s*[that arises from taking \(k\rightarrow \infty \) in (2.1)]. Then, using \(\rho _s=1-\gamma /\sqrt{s}\), with

### Proposition 5.1

### Proof

### 5.2 Asymptotic series for \(P(Q>N)\) as \(N\rightarrow \infty \)

*R*of the integration contour to values \(R_M\) between \(|Z({\pm }M)|\) and \(|Z({\pm }(M+1))|\) when \(M=1,2,...\) is fixed. Here it must be assumed that

*s*is so large that \(|Z_k|\) increases in \(k=0,1,...,M+1\). Then, the poles of

*Q*(

*z*) at \(z=Z_{\pm k}\), \(k=0,1,...,M \), are inside \(|z|=R_M\), and we get

We now need the following result.

### Lemma 5.2

### Proof

This follows from the appendix with a similar argument to the proof of Lemma 3.3. \(\square \)

*k*, see Lemma 5.2,

### Proposition 5.3

*k*and large

*s*. To that end, we conduct the dedicated saddle point analysis for \(I(Z_k)\). We have for \(|Z|\ge Z_0\), \(\mathrm{Re}(Z)>z_\mathrm{sp}\),

*z*(

*v*) as in (4.21) and defined implicitly by \(g(z(v))=g(z_\mathrm{sp})-\frac{1}{2} v^2g''(z_\mathrm{sp})\). We then find, by using \(z(v)=z_\mathrm{sp}+iv+O(v^2)\) and \(z'(v)=i+O(v)\), that

*t*and \(-t\) for \(t\ge 0\), we get the following result.

### Proposition 5.4

### Proposition 5.5

*J*is given in (5.19).

### Theorem 5.6

*M*and \(M-1\) is \(O(|Z_{M-1}/Z_M|^N)\).

### Proof

This follows from (5.10), in which the integral is \(o(|Z_M|^{-N})\) and the term with \(k=M\) is \(O(|Z_M|^{-N})\), while the reciprocal of the term with \(k=M-1\) is \(O(|Z_{M-1}|^{-N})\) by Proposition 5.5. In the consideration of the terms with \(k=M-1,M \), it is tacitly assumed that *s* is so large that \(|Z_k|\), \(k=0,1,...,M\) is a strictly increasing sequence. \(\square \)

## Notes

### Acknowledgments

This work was financially supported by The Netherlands Organization for Scientific Research (NWO) and by an ERC Starting Grant.

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