Time-Dependant Cases

Consider the MCS when the call rate $\lambda=\lambda(t)$, and the number of servers $m=m(t)$ and waiting places $r=r(t)$ all depend on time. The main difference from the previous chapter is that all the parameters depend on time therefore all the results should be represented as functions of time. Here are the expressions similar to those in the previous chapter modified by including the time factor.

• denote by $t_n$ the "arriving" time of the $n$th call, then $t_{n+1}=t_n+\xi_n(\lambda(t)),\ \xi(\lambda(t))= - 1/\lambda(t)\ log(1-\eta)$ where $\eta$ is a random number uniform in $[0,1]$ and $\lambda$ is the average number of calls arriving in a time unit at the moment $t$
• denote by $\tau_n$ the "departing" time of $n$th call, then $\tau_{n+1}=\tau_n+\xi_n(\mu_n),\ \xi(\mu_n) = - 1/\mu_n\ log(1-\eta)$ where $\mu_n$ is the average number of calls which can be served in a time unit by all the $1 \le m_n \le m(\tau_n)$ servers operating at the moment $\tau_n$, thus $\mu_n= m_n \ \mu$ and $\mu$ is the average number of calls served by single server in unit time
• the number of all the calls in the system at the moment $t_{n+1}$ including those in $m(t_{n+1})$ servers and $r(t_{n+1})$ waiting places
  

  $$I_{n+1} = \cases {I_n-1,$$ (100)

  
  
  where $I_n$ is the number of calls in the system at the moment $t_n$
• the number of servers $m_n$ operating at the moment $\tau_n$
  

  $$m_n=\min (I_l,m(\tau_n)),\ t_l \le \tau_n \le t_{l+1}$$ (101)

  
  

• the number of calls lost up to the moment $t_{n+1}$
  

  $$L_{n+1} = \cases {L_n+1,$$ (102)

  
  
  where $L_n$ is the number of calls lost up to the the moment $t_n$
• the waiting time counter
  

  $$T_{n+1} =T_n+\delta_{n},$$ (103)

  $$\delta_{n}=\tau_{n}-t_{n}$$ (104)

  
  

• the counter of the call numbers $q^k$ that waited less than $t^k,\ t^{k+1} > t^k,\ k=1,...,K$
  

  $$q_{n+1}^k = \cases {\min(q_n^k+1,m(t_{n+1})+r(t_{n+1})),$$ (105)

  
  

• distribution of the call numbers $q^k$ that waited less than $t^k,\ t^{k+1} > t^k,\ k=1,...,K$
  

  $$Q(t^k)=1/N\ q_{N}^k$$ (106)

  
  

• optimization of number $m(t)$ as a function of $t$ can be formulated as the stochastic scheduling problem solution of which is very difficult in general, thus later on we shall consider some simple cases as illustrations
• defining the difference between the steady-state and dynamic solutions the average results are compared with the corresponding steady-state explicit solutions including the waiting time distribution functions $F_{\tau}^{m,r}(t)= P^{m,r}\{\tau <t\}$, where $P^{m,r}\{\tau <t\}$ is the probability that the waiting time $\tau$ will be less than $t$
• the iteration number $N$ is defined by experimentation