Simple Example

One may start by simple example

$$\lambda(t)=a+b\ sin(\phi+ 2 \pi \omega t)$$ (107)

Here $\omega=1/24$ and $\phi = 8/24$ where $t$ is in hours, $a$ is average number of calls in 24 hours, and $b$ is amplitude of daily variation of calls and

$$m(t) = \cases {m^0,$$ (108)

and $r(t)=r$

Consider optimization of server numbers. We optimize the numbers $m^0$ and $m^1$ of two different servers considering two periods of time

• there are two optimization parameters $m^0$ and $m^1$ restricted by bounds $m_{min} \le m^0 \le m_{max}^0$ and $m_{min}^1 \le m^1 \le m_{max}^1$
• one may obtain the optimal pair by direct comparison of all the feasible pairs
  

  $$m*=(m^{*0},m^{*1})=arg \min_{m^0,m^1} C(c^0,c^1,m^0,m^1,r)$$ (109)

  
  

• the total system cost is defined as
  

  $$C(c^0,c^1,m^0,m^1,r)=c^0 m^0 + c^1 m^1 + c_t T(m^0,m^1,r,\mu, a, b, \phi, \omega)$$

  $$+c_p P_{m^0,m^1,r}(\mu, a, b, \phi, \omega).$$ (110)