15.3 Approximations of the Ruin Probability in Infinite Time

When the claim size distribution is exponential (or closely related to it), simple analytic results for the ruin probability in infinite time exist, see Section 15.2. For more general claim amount distributions, e.g. heavy-tailed, the Laplace transform technique does not work and one needs some estimates. In this section, we present 12 different well-known and not so well-known approximations. We illustrate them on a common claim size distribution example, namely the mixture of two exponentials claims with $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$ and $ a=0.0584$ (see Chapter 13). Numerical comparison of the approximations is given in Section 15.4.


15.3.1 Cramér-Lundberg Approximation

Cramér-Lundberg's asymptotic ruin formula for $ \psi(u)$ for large $ u$ is given by

$\displaystyle \psi_{CL} (u) = Ce^{-Ru},$ (15.13)

where $ C=\theta \mu/\left\{M_{X}'(R)-\mu (1+\theta )\right\}.$ For the proof we refer to Grandell (1991). The classical Cramér-Lundberg approximation yields quite accurate results, however we must remember that it requires the adjustment coefficient to exist, therefore merely the light-tailed distributions can be taken into consideration. For exponentially distributed claims, the formula (15.13) yields the exact result.

In Table 15.5 the Cramér-Lundberg approximation for mixture of two exponentials claims with $ \beta_1$, $ \beta_2$, $ a$ and the relative safety loading $ \theta=30\%$ with respect to the initial capital $ u$ is given. We see that the Cramér-Lundberg approximation underestimates the ruin probability. Nevertheless, the results coincide quite closely with the exact values shown by Table 15.4. When the initial capital is zero, the relative error is the biggest and exceeds $ 13\%$.


Table 15.5: The Cramér-Lundberg approximation for mixture of two exponentials claims with $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$, $ a=0.0584$ and $ \theta =0.3$ ($ u$ in USD billion).
$ u$ 0 $ 1$ $ 5$ $ 10$ $ 20$ $ 50$
             
$ \psi_{CL} (u)$ 0.663843 0.587260 0.359660 0.194858 0.057197 0.001447
             

24398 STFruin05.xpl


15.3.2 Exponential Approximation

This approximation was proposed and derived by De Vylder (1996). It requires the first three moments to be finite.

$\displaystyle \psi_{E}(u)= \exp\left\{-1-\frac{2\mu\theta u-\mu^{(2)}}{\sqrt{(\mu^{(2)})^{2}+(4/3)\theta\mu\mu^{(3)}}} \right\}.$ (15.14)

Table 15.6 shows the results of the exponential approximation for mixture of two exponentials claims with $ \beta_1$, $ \beta_2$, $ a$ and the relative safety loading $ \theta=30\%$ with respect to the initial capital $ u$. Comparing them with the exact values presented in Table 15.4 we see that the exponential approximation works not bad in the studied case. When the initial capital is USD $ 50$ billion, the relative error is the biggest and reaches $ 24\%$.


Table 15.6: The exponential approximation for mixture of two exponentials claims with $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$, $ a=0.0584$ and $ \theta =0.3$ ($ u$ in USD billion).
$ u$ 0 $ 1$ $ 5$ $ 10$ $ 20$ $ 50$
             
$ \psi_E(u)$ 0.747418 0.656048 0.389424 0.202900 0.055081 0.001102
             

24490 STFruin06.xpl


15.3.3 Lundberg Approximation

The following formula, called the Lundberg approximation, comes from Grandell (2000). It requires the first three moments to be finite.

$\displaystyle \psi_{L}(u)= \left \{ 1+ \left(\theta u-\frac{\mu^{(2)}}{2\mu} \r...
...} {3(\mu^{(2)})^{3}}\right\} \exp \left(\frac{-2\mu\theta u}{\mu^{(2)}}\right).$ (15.15)

In Table 15.7 the Lundberg approximation for mixture of two exponentials claims with $ \beta_1$, $ \beta_2$, $ a$ and the relative safety loading $ \theta=30\%$ with respect to the initial capital $ u$ is given. We see that the Lundberg approximation works worse than the exponential one. When the initial capital is USD $ 50$ billion, the relative error exceeds $ 60\%$.


Table 15.7: The Lundberg approximation for mixture of two exponentials claims with $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$, $ a=0.0584$ and $ \theta =0.3$ ($ u$ in USD billion).
$ u$ 0 $ 1$ $ 5$ $ 10$ $ 20$ $ 50$
             
$ \psi_L (u)$ 0.504967 0.495882 0.382790 0.224942 0.058739 0.000513
             

24584 STFruin07.xpl


15.3.4 Beekman-Bowers Approximation

The Beekman-Bowers approximation uses the following representation of the ruin probability:

$\displaystyle \psi(u)=\textrm{P}(L>u)=\textrm{P}(L>0)\textrm{P}(L>u\vert L>0).$ (15.16)

The idea of the approximation is to replace the conditional probability $ 1-\textrm{P}(L>u\vert L>0)$ with a gamma distribution function $ G(u)$ by fitting first two moments (Grandell; 2000). This leads to:

$\displaystyle \psi_{BB}(u)=\frac{1}{1+\theta}\left\{ 1-G(u) \right\},$ (15.17)

where the parameters $ \alpha,\;\beta$ of $ G$ are given by

$\displaystyle \alpha=\left\{1+\left(\frac{4\mu\mu^{(3)}}{3(\mu^{(2)})^{2}}-1\ri...
...{(2)}+\left (\frac{4\mu\mu^{(3)}}{3\mu^{(2)}}-\mu^{(2)}\right )\theta\right\}.
$

The Beekman-Bowers approximation gives rather accurate results, see Burnecki, Mista, and Weron (2004). In the exponential case it becomes the exact formula. It can be used only for distributions with finite first three moments.


Table 15.8: The Beekman-Bowers approximation for mixture of two exponentials claims with $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$, $ a=0.0584$ and $ \theta =0.3$ ($ u$ in USD billion).
$ u$ 0 $ 1$ $ 5$ $ 10$ $ 20$ $ 50$
             
$ \psi_{BB}(u)$ 0.769231 0.624902 0.352177 0.186582 0.056260 0.001810
             

24690 STFruin08.xpl

Table 15.8 shows the results of the Beekman-Bowers approximation for mixture of two exponentials claims with $ \beta_1$, $ \beta_2$, $ a$ and the relative safety loading $ \theta=30\%$ with respect to the initial capital $ u$. The results justify the thesis the approximation yields quite accurate results but when the initial capital is USD $ 50$ billion, the relative error is unacceptable, reaching $ 25\%$, cf. the exact values in Table 15.4.


15.3.5 Renyi Approximation

The Renyi approximation (Grandell; 2000), may be derived from (20.5.4) when we replace the gamma distribution function $ G$ with an exponential one, matching only the first moment. Hence, it can be regarded as a simplified version of the Beekman-Bowers approximation. It requires the first two moments to be finite.

$\displaystyle \psi_{R}(u)= \frac{1}{1+\theta}\exp\left\{-\frac{2\mu\theta u}{\mu^{(2)}(1+\theta)}\right\}.$ (15.18)

In Table 15.9 the Renyi approximation for mixture of two exponentials claims with $ \beta_1$, $ \beta_2$, $ a$ and the relative safety loading $ \theta=30\%$ with respect to the initial capital $ u$ is given. We see that the results compared with the exact values presented in Table 15.4 are quite accurate. The accuracy ot the approximation is similar to the Beekman-Bowers approximation but when the initial capital is USD $ 50$ billion, the relative error exceeds $ 50\%$.


Table 15.9: The Renyi approximation for mixture of two exponentials claims with $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$, $ a=0.0584$ and $ \theta =0.3$ ($ u$ in USD billion).
$ u$ 0 $ 1$ $ 5$ $ 10$ $ 20$ $ 50$
             
$ \psi_R(u) $ 0.769231 0.667738 0.379145 0.186876 0.045400 0.000651
             

24792 STFruin09.xpl


15.3.6 De Vylder Approximation

The idea of this approximation is to replace the claim surplus process $ S_t$ with the claim surplus process $ \bar{S_t}$ with exponentially distributed claims such that the three moments of the processes coincide, namely $ \mathop{\textrm{E}}(S^k_t)=\mathop{\textrm{E}}(\bar{S}_t^k)$ for $ k=1,2,3$, see De Vylder (1978). The process $ \bar{S}_t$ is determined by the three parameters $ (\bar{\lambda}$, $ \bar{\theta}, \bar{\beta})$. Thus the parameters must satisfy:

$\displaystyle \qquad\bar{\lambda}=\frac{9\lambda\mu^{(2)^{3}}}{2\mu^{(3)^{2}}},...
...^{2}}}\theta, \qquad {\rm and} \qquad\bar{\beta}=\frac{3\mu^{(2)}}{\mu^{(3)}}.
$

Then De Vylder's approximation is given by:

$\displaystyle \psi_{DV}(u)=\frac{1}{1+\bar{\theta}}\exp\left(-\frac{\bar{\theta}\bar{\beta}u}{1+\bar{\theta}}\right).$ (15.19)

Obviously, in the exponential case the method gives the exact result. For other claim amount distributions, in order to apply the approximation, the first three moments have to exist.

Table 15.10 shows the results of the De Vylder approximation for mixture of two exponentials claims with $ \beta_1$, $ \beta_2$, $ a$ and the relative safety loading $ \theta=30\%$ with respect to the initial capital $ u$. The approximation gives surprisingly good results. In the considered case the relative error is the biggest when the initial capital is zero and amounts to about $ 13\%$, cf. Table 15.4.


Table 15.10: The De Vylder approximation for mixture of two exponentials claims with $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$, $ a=0.0584$ and $ \theta =0.3$ ($ u$ in USD billion).
$ u$ 0 $ 1$ $ 5$ $ 10$ $ 20$ $ 50$
             
$ \psi_{DV}(u)$ 0.668881 0.591446 0.361560 0.195439 0.057105 0.001424
             

24923 STFruin10.xpl


15.3.7 4-moment Gamma De Vylder Approximation

The 4-moment gamma De Vylder approximation, proposed by Burnecki, Mista, and Weron (2003), is based on De Vylder's idea to replace the claim surplus process $ S_t$ with another one $ \bar{S}_t$ for which the expression for $ \psi(u)$ is explicit. This time we calculate the parameters of the new process with gamma distributed claims and apply the exact formula (15.10) for the ruin probability. Let us note that the claim surplus process $ \bar{S_t}$ with gamma claims is determined by the four parameters $ (\bar{\lambda}, \bar{\theta}, \bar{\mu}, \bar{\mu}^{(2)})$, so we have to match the four moments of $ S_t$ and $ \bar{S_t}$. We also need to assume that $ \mu^{(2)}\mu^{(4)}<\frac{3}{2}(\mu^{(3)})^{2}$ to ensure that $ \bar{\mu},\bar{\mu}^{(2)}>0$ and $ \bar{\mu}^{(2)}>\bar{\mu}^{2}$, which is true for the gamma distribution. Then

\begin{displaymath}
\begin{array}{lcl}
\bar{\lambda} &=& \frac{\lambda(\mu^{(3)...
...(\mu^{(3)})^{2}\right\}}{(\mu^{(2)}\mu^{(3)})^{2}}.
\end{array}\end{displaymath}

When this assumption can not be fulfilled, the simpler case leads to

$\displaystyle \bar{\lambda}=\frac{2\lambda(\mu^{(2)})^{2}}{\mu(\mu^{(3)}+\mu^{(...
...r{\mu}=\mu,\;\; \bar{\mu}^{(2)}=\frac{\mu(\mu^{(3)}+\mu^{(2)}\mu)}{2\mu^{(2)}}.$    

All in all, the 4-moment gamma De Vylder approximation is given by

$\displaystyle \psi_{4MGDV} (u) = \frac{\bar{\theta}(1-\frac{R}{\bar{\alpha}})\e...
...\alpha}})} + \frac{\bar{\alpha}\bar{\theta} \sin(\bar{\alpha}\pi)}{\pi}\cdot I,$ (15.20)

where

$\displaystyle I = \int_{0}^{\infty}\frac{x^{\bar{\alpha}}\exp\{-(x+1)\bar{\beta...
...a})(x+1)\right\}-\cos(\bar{\alpha}\pi)\right ]^{2}+\sin^{2}(\bar{\alpha}\pi)},
$

and $ \bar{\alpha}=\bar{\mu}^{2}/\left(\bar{\mu}^{(2)}-\bar{\mu}^{2}\right)$, $ \bar{\beta}=\bar{\mu}/\left(\bar{\mu}^{(2)}-\bar{\mu}^{2}\right)$.

In the exponential and gamma case this method gives the exact result. For other claim distributions in order to apply the approximation, the first four (or three in the simpler case) moments have to exist. Burnecki, Mista, and Weron (2003) showed numerically that the method gives a slight correction to the De Vylder approximation, which is often regarded as the best among ``simple'' approximations.

In Table 15.11 the 4-moment gamma De Vylder approximation for mixture of two exponentials claims with $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$, $ a=0.0584$ (see Chapter 13) and the relative safety loading $ \theta=30\%$ with respect to the initial capital $ u$ is given. The most striking impression of Table 15.11 is certainly the extremely good accuracy of the simple 4-moment gamma De Vylder approximation for reasonable choices of the initial capital $ u$. The relative error with respect to the exact values presented in Table 15.4 is the biggest for $ u=0$ and equals $ 11\%$.


Table 15.11: The 4-moment gamma De Vylder approximation for mixture of two exponentials claims with $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$, $ a=0.0584$ and $ \theta =0.3$ ($ u$ in USD billion).
$ u$ 0 $ 1$ $ 5$ $ 10$ $ 20$ $ 50$
$ \psi_{4MGDV}(u)$ 0.683946 0.595457 0.359879 0.194589 0.057150 0.001450

25182 STFruin11.xpl


15.3.8 Heavy Traffic Approximation

The term ``heavy traffic'' comes from queuing theory. In risk theory it means that, on the average, the premiums exceed only slightly the expected claims. It implies that the relative safety loading $ \theta$ is positive and small. Asmussen (2000) suggests the following approximation.

$\displaystyle \psi_{HT}(u)= \exp\left(-\frac{2\theta\mu u}{\mu^{(2)}}\right).$ (15.21)

This method requires the existence of the first two moments of the claim size distribution. Numerical evidence shows that the approximation is reasonable for the relative safety loading being $ 10-20\%$ and $ u$ being small or moderate, while the approximation may be far off for large $ u$. We also note that the approximation given by (15.21) is also known as the diffusion approximation and is further analysed and generalised to the stable case in Chapter 16, see also Furrer, Michna, and Weron (1997).

Table 15.12 shows the results of the heavy traffic approximation for mixture of two exponentials claims with $ \beta_1$, $ \beta_2$, $ a$ and the relative safety loading $ \theta=30\%$ with respect to the initial capital $ u$. It is clear that the accuracy of the approximation in the considered case is extremely poor. When the initial capital is USD $ 50$ billion, the relative error reaches $ 93\%$, cf. Table 15.4.


Table 15.12: The heavy traffic approximation for mixture of two exponentials claims with $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$, $ a=0.0584$ and $ \theta =0.3$ ($ u$ in USD billion).
$ u$ 0 $ 1$ $ 5$ $ 10$ $ 20$ $ 50$
             
$ \psi_{HT}(u)$ 1.000000 0.831983 0.398633 0.158908 0.025252 0.000101
             

25312 STFruin12.xpl


15.3.9 Light Traffic Approximation

As for heavy traffic, the term ``light traffic'' comes from queuing theory, but has an obvious interpretation also in risk theory, namely, on the average, the premiums are much larger than the expected claims, or in other words, claims appear less frequently than expected. It implies that the relative safety loading $ \theta$ is positive and large. We may obtain the following asymptotic formula.

$\displaystyle \psi_{LT}(u)=\frac{1}{(1+\theta)\mu}\int_{u}^{\infty}\bar{F}_X(x)dx.$ (15.22)

In risk theory heavy traffic is most often argued to be the typical case rather than light traffic. However, light traffic is of some interest as a complement to heavy traffic, as well as it is needed for the interpolation approximation to be studied in the next point. It is worth noticing that this method gives accurate results merely for huge values of the relative safety loading, see Asmussen (2000).

In Table 15.13 the light traffic approximation for mixture of two exponentials claims with $ \beta_1$, $ \beta_2$, $ a$ and the relative safety loading $ \theta=30\%$ with respect to the initial capital $ u$ is given. The results are even worse than in the heavy case, only for moderate $ u$ the situation is better. The relative error dramatically increases with the initial capital.


Table 15.13: The light traffic approximation for mixture of two exponentials claims with $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$, $ a=0.0584$ and $ \theta =0.3$ ($ u$ in USD billion).
$ u$ 0 $ 1$ $ 5$ $ 10$ $ 20$ $ 50$
             
$ \psi_{LT}(u)$ 0.769231 0.303545 0.072163 0.011988 0.000331 0.000000
             

25413 STFruin13.xpl


15.3.10 Heavy-light Traffic Approximation

The crude idea of this approximation is to combine the heavy and light approximations (Asmussen; 2000):

$\displaystyle \psi_{HLT}(u)=\frac{\theta}{1+\theta}\psi_{LT}\left(\frac{\theta u}{1+\theta}\right) +\frac{1}{(1+\theta)^{2}}\psi_{HT}(u).$ (15.23)

The particular features of this approximation is that it is exact for the exponential distribution and asymptotically correct both in light and heavy traffic.

Table 15.14 shows the results of the heavy-light traffic approximation for mixture of two exponentials claims with $ \beta_1$, $ \beta_2$, $ a$ and the relative safety loading $ \theta=30\%$ with respect to the initial capital $ u$. Comparing the results with Table 15.12 (heavy traffic), Table 15.13 (light traffic) and the exact values given in Table 15.4 we see that the interpolation is promising. In the considered case the relative error is the biggest when the initial capital is USD $ 20$ billion and is over $ 40\%$, but usually the error is acceptable.


Table 15.14: The heavy-light traffic approximation for mixture of two exponentials claims with $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$, $ a=0.0584$ and $ \theta =0.3$ ($ u$ in USD billion).
$ u$ 0 $ 1$ $ 5$ $ 10$ $ 20$ $ 50$
             
$ \psi_{HLT}(u)$ 0.769231 0.598231 0.302136 0.137806 0.034061 0.001652
             

25508 STFruin14.xpl


15.3.11 Subexponential Approximation

First, let us introduce the class of subexponential distributions $ \mathcal{S}$ (Embrechts, Klüppelberg, and Mikosch; 1997), namely

$\displaystyle \mathcal{S}= \left\{ F:\lim_{x\rightarrow \infty}\frac{\overline{F^{\ast 2}}(x)}{\bar{F}(x)}=2\right \}.$ (15.24)

Here $ F^{\ast 2}(x)$ is the convolution square. In terms of random variables (15.24) means $ \textrm{P}(X_1+X_2>x)\sim 2P(X_1>x)$, $ x\rightarrow\infty$, where $ X_1$, $ X_2$ are independent random variables with distribution $ F$.

The class contains log-normal and Weibull (for $ \tau <1$) distributions. Moreover, all distributions with a regularly varying tail (e.g. Pareto and Burr distributions) are subexponential. For subexponential distributions we can formulate the following approximation of the ruin probability. If $ F \in \mathcal{S}$, then the asymptotic formula for large $ u$ is given by

$\displaystyle \psi_{S} (u) = \frac{1}{\theta \mu}\left(\mu-\int_{0}^{u}\bar{F}(x)dx \right),$ (15.25)

see Asmussen (2000).

The approximation is considered to be inaccurate. The problem is a very slow rate of convergence as $ u\rightarrow \infty$. Even though the approximation is asymptotically correct in the tail, one may have to go out to values of $ \psi(u)$ which are unrealistically small before the fit is reasonable. However, we will show in Section 15.4 that it is not always the case.

As the mixture of exponentials does not belong to the subexponential class we do not present a numerical example like in all previously discussed approximations.


15.3.12 Computer Approximation via the Pollaczek-Khinchin Formula

One can use the Pollaczek-Khinchin formula (15.8) to derive explicit closed form solutions for claim amount distributions presented in Section 15.2, see Panjer and Willmot (1992). For other distributions studied here, in order to calculate the ruin probability, the Monte Carlo method can be applied to (15.1) and (15.7). The main problem is to simulate random variables from the density $ f_{L_1}(x)$. Only four of the considered distributions lead to a known density: (i) for exponential claims, $ f_{L_1}(x)$ is the density of the same exponential distribution, (ii) for a mixture of exponentials claims, $ f_{L_1}(x)$ is the density of the mixture of exponential distribution with the weights $ \left(\frac{a_{1}}{\beta_{1}}/\left\{\sum_{i=1}^{n}\left(\frac{a_{i}}{\beta_{i}}\right)\right\},
\cdots,\right.$ $ \left.\frac{a_{n}}{\beta_{n}}/\left\{\sum_{i=1}^{n}\left(\frac{a_{i}}{\beta_{i}}\right)\right\}\right)$, (iii) for Pareto claims, $ f_{L_1}(x)$ is the density of the Pareto distribution with the parameters $ \alpha-1$ and $ \lambda $, (iv) for Burr claims, $ f_{L_1}(x)$ is the density of the transformed beta distribution.


Table 15.15: The Pollaczek-Khinchin approximation for mixture of two exponentials claims with $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$, $ a=0.0584$ and $ \theta =0.3$ ($ u$ in USD billion).
$ u$ 0 $ 1$ $ 5$ $ 10$ $ 20$ $ 50$
             
$ \psi_{PK}(u)$ 0.769209 0.587917 0.359705 0.194822 0.057173 0.001445
             

25700 STFruin15.xpl

For other distributions studied here we use formula (15.6) and controlled numerical integration to generate random variables $ L_k$ (except for the Weibull distribution, $ f_{L_1}(x)$ does not even have a closed form). We note that the methodology based on the Pollaczek-Khinchin formula works for all considered claim distributions.

The computer approximation via the Pollaczek-Khinchin formula will be called in short the Pollaczek-Khinchin approximation. Burnecki, Mista, and Weron (2004) showed that the approximation can be chosen as the reference method for calculating the ruin probability in infinite time, see also Table 15.15 where the results of the Pollaczek-Khinchin approximation are presented for mixture of two exponentials claims with $ \beta_1$, $ \beta_2$, $ a$ and the relative safety loading $ \theta=30\%$ with respect to the initial capital $ u$. For the Monte Carlo method purposes we generated 100 blocks of 500000 simulations.


15.3.13 Summary of the Approximations

Table 15.16 shows which approximation can be used for a particular choice of a claim size distribution. Moreover, the necessary assumptions on the distribution parameters are presented.


Table 15.16: Survey of approximations with an indication when they can be applied
$ \qquad$Distrib. Exp. Gamma Wei- Mix. Log- Pareto Burr
Method bull Exp. normal
Cramér-Lundberg + + - + - - -
Exponential + + + + + $ \alpha>3$ $ \alpha\tau>3$
Lundberg + + + + + $ \alpha>3$ $ \alpha\tau>3$
Beekman- + + + + + $ \alpha>3$ $ \alpha\tau>3$
$ \quad$-Bowers
Renyi + + + + + $ \alpha>2$ $ \alpha\tau>2$
De Vylder + + + + + $ \alpha>3$ $ \alpha\tau>3$
4M Gamma + + + + + $ \alpha>3$ $ \alpha\tau>3$
$ \quad$De Vylder
Heavy Traffic + + + + + $ \alpha>2$ $ \alpha\tau>2$
Light Traffic + + + + + + +
Heavy-Light + + + + + $ \alpha>2$ $ \alpha\tau>2$
$ \quad$Traffic
Subexponential - - 0$ <$$ \tau $$ <$1 - + + +
Pollaczek- + + + + + + +
$ \quad$-Khinchin