On Server Efficiency

# On Server Efficiency

## Abstract

Editor’s note: This paper comprises the second chapter of the PhD dissertation by Rachel Traylor. Cha and Lee defined a mathematical notion of server performance by measuring efficiency $\psi$ defined as the long run average number of jobs completed per unit time. The service time distribution heavily influences the shape of the server efficiency as a function of a constant arrival rate $\lambda$. Various classes of distributions are studied in order to find sufficient conditions for the existence of a single maximum. The existence of a maximum allows for simple binary control policies to handle traffic and optimize performance.

## Introduction, Motivation, and Background

Cha and Lee  studied the reliability of a single server under a constant stress workload, and also defined a notion of server efficiency $\psi$ for a given intensity $\lambda(t)$ as the long-run average number of jobs completed per unit time as a way to measure server performance. With the number of jobs completed as $M$, the efficiency is defined

$$\psi := \lim\limits_{t \to \infty}\frac{E[M(t)]}{t}$$

Upon breakdown and rebooting, the server is assumed to be ‘as good as new’, in that performance of the server does not degrade during subsequent reboots. In addition, the model assumes the arrival process after reboot, denoted $\{N^{*}(t), t \geq 0\}$, is a nonhomogenous Poisson process with the same intensity function $\lambda(t)$ as before, and that $\{N^{*}(t), t \geq 0\}$ is independent of the arrival process before reboot. In a practical setting, this model assumes no ‘bottlenecking’ of arrivals occurs in the queue during server downtime that would cause an initial flood to the rebooted server. In addition, the reboot time is assumed to follow a continuous distribution $H(t)$ with expected value $\nu$. This process is a renewal reward process, with the renewal $\{R_{n}\} = \{M_{n}\}$, the number of jobs completed. The length of a renewal cycle is $Y_{n} + H_{n}$, where $Y_{n}$ is the length of time the server was operational, and $H_{n}$ is the time to reboot after a server crash. Then, by ,

$$\psi = \frac{E[M]}{E[Y]+ \nu}$$

where $M$ is the number of jobs completed in a particular renewal cycle, $\nu$ is the mean time to reboot of the server, and $Y$ is the length of a particular renewal cycle. Then, using the definition of $\psi$ the following closed form of the efficiency of a server under all assumptions of Cha and Lee’s model is derived.

Theorem 1 (Server Efficiency under Cha/Lee)
Suppose $\{N(t), t \geq 0\}$ is a nonhomogenous Poisson process with intensity $\lambda(t)\geq 0$. Then the efficiency is given by

\begin{aligned}\psi&=\frac{1}{\int_{0}^{\infty}S_{Y}(t)dt + \nu}\left[\exp\left(-\int_{0}^{t}r_{0}(x)dx-\int_{0}^{t}\lambda(x)dx + a(t) + b(t)\right)\right.\\&\qquad\qquad\left.\times\left(r_{0}(t)a(t)+\eta a(t)b(t) \right)\right]\end{aligned} where $a(t) = \int_{0}^{t}e^{-\eta v}g_{W}(v)m(t-v)dv$, $b(t) = \int_{0}^{t}e^{-\eta(t-r)}\bar{G}_{W}(t-r)\lambda(r)dr$, $\bar{G}_{W}(x) = 1-\int_{0}^{x}g_{W}(s)ds$, and $m(x) = \int_{0}^{x}\lambda(s)ds$.

### Numerical Example and Control Policies

As an illustrative example, Cha and Lee considered the case when $\lambda(t) \equiv \lambda$,
$r_{0}(t) \equiv r_{0} = 0.2$, $\eta = 0.01$, $\nu = 1$, and $g_{W}(w) = we^{-w^{2}/2}$ (the PDF of the Rayleigh distribution). As shown in Figure 1, there exists a $\lambda^{*}$ such that $\psi(\lambda)$ is maximized. Thus one may implement the obvious optimal control policy for server control to avoid server overload:

(1) If the real time arrival rate $\lambda < \lambda^{*}$, do not interfere with arrivals.

(2) If $\lambda \geq \lambda^{*}$, facilitate some appropriate measure of interference.

Examples of interference for a web server in particular include rejection of incoming requests or possible re-routing. Cha and Lee give an interference policy of rejection with probability $1-\frac{\lambda^{*}}{\lambda}$.

The Rayleigh distribution used in Figure 1 has applications in physics, typically when the magnitude of a vector is related to its directional components. It is a special case of the Weibull distribution which is widely used in survival analysis, failure analysis, weather forecasting, and communications. These distributions are not typically used to model service times. The exponential distribution is the most common due to its memoryless properties, followed by the Erlang and uniform distributions.

The efficiency, $\psi$, under the Rayleigh distribution example in Figure 1 shows the existence of a $0 < \lambda^{*} < \infty$ such that $\psi(\lambda)$ is maximized at $\lambda^{*}$. This useful feature of $\psi(\lambda)$ in this case allows for the implementation of a simple control policy for arrivals to the server to prevent overload, given above.

Numerical simulations under a variety of possible distribution classes, including convex, concave, exponential, uniform, and Erlang suggest that the mathematical properties of $\psi$ are heavily influenced by the choice and characteristics of service time distribution $g_{W}(w)$. In particular, it is of interest to seek sufficient conditions of $g_{W}(w)$ that will guarantee the existence of a $\lambda^{*}$ that maximizes $\psi$. This is done for the uniform, compact support, and Erlang classes. Furthermore, it is shown under certain conditions, not only does the server efficiency lack a maximum, but $\psi$ increases without bound. This is not representative of real server behavior, and thus of mathematical and practical interest to note for further modeling.

## Efficiency of a Server under Uniform Service Life Distribution

Suppose $\lambda(x) \equiv \lambda$, and suppose $r_{0}(x) \equiv r_{0} = \max_{x \in (0,\infty)}r_{0}(x)$. The efficiency $\psi$ is given for constant $\eta$ and $\lambda$ here:
$\psi(\lambda) = \frac{1}{\int_{0}^{\infty}S_{Y}(t)dt + \nu}\left[\int_{0}^{\infty}\exp\left(-r_{0}t-\lambda t + a(t)+b(t)\right)(r_{0}+b(t))a(t)dt\right]$

where $S_{Y}(t)$ is the survival function of the node, $a(t) = \int_{0}^{t}e^{-\eta v}g(v)(t-v)dv$, $b(t) = \int_{0}^{t}e^{-\eta(t-r)}\bar{G}(t-r)dr$,
$g(v)$ is the pdf of the service time distribution, and $\bar{G}(x) = 1-\int_{0}^{x}g(s)ds$.

The following theorem gives sufficient conditions for the uniform distribution and $\eta$ that guarantee the existence of a finite maximum efficiency.

Theorem 2(Efficiency under Uniform Service Distribution)
Suppose the service life distribution is given by Uniform($c,d$) for some $0. Then if
$\sigma > \frac{ce^{-c\eta}}{\sqrt{12}\phi(-\eta)(1+\eta(c+d))+c\eta - 1}$ where $\phi(-\eta)$ the standard deviation of the service life $W$, $\psi(\lambda)$ has a maximum on $(0,\infty)$ is the moment generating function of a uniform distribution evaluated at $-\eta$.

Numerical simulations suggest that $\psi$ increases without bound for $c=0, d>1$. The following lemma proves this fact.

Lemma
Suppose the service life distribution is given by Uniform(0,d), with $d>1$. Then $\psi$ increases without bound.

This is worth discussing here. It makes no sense whatsoever for the efficiency of a server to increase forever as the arrival rate increases. So what’s happening here? Notice that if the uniform distribution include 0 as an endpoint, then there is a positive probability that the service time for a job will be exactly 0 (in whatever time units you like). This is impossible in reality. A small service time is possible, but it’s still strictly greater than 0. What we see here is that using distributions with positive support at 0 causes issues in the efficiency function; it’s not the fault of the definition of efficiency, but rather a consequence of using distributions that cannot mirror reality. The next section explores this further with a more broad class of distributions.

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