Assumptions, Notations, and Objectives

We regard call center as a queuing system including the features for call rate prediction and service optimization. We assume that

• incoming calls are united into one stream with the call rate $\lambda$, a call rate is the average number of calls in a time unit
• there are $m$ servers with the same service rate $\mu$, the service rate is the average number of calls which can be served in a time unit,
• each server can serve any call, one at a time
• calls are Poisson with the rate $\lambda$
• service times are exponential with the rate $\mu > 1/m\ \lambda$, in such a case
  

  $$\lambda=1/T_{\lambda}$$ (61)

  $$\mu=1/T_{\mu}$$ (62)

  
  
  where $T_{\lambda}$ is an average time between the calls and $T_{\mu}$ is an average service time.
• arriving calls enter the first available server, if all the servers are busy then the call takes a free waiting place
• there are $r$ waiting places , that means that a call waits if there are not more than $r$ other calls waiting, otherwise the call disappears, the waiting places are common for all the calls
• the system is stationary, meaning that the parameters $\mu$ and $\lambda$ are constant.
• $c_m$ is the server running cost ( $ per time unit)
• $c_t$ is the customer time cost ( $ per time unit)
• $c_p$ is the lost customer cost ( $ per lost call)
• the optimal number of servers $m=m(c,r)$ minimizes the total cost $C(c,m)$ per time unit, including the server running cost, the customer time cost, and the lost customer cost at fixed parameters $r$ and $c=(c_s,c_t,c_p)$
• results are presented as 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 $t$ at fixed number of servers $m$ and waiting places $r$.
• a family of functions $F_{\tau}^{c,r}(t)$ is defined for different parameters $c,r$ and presented in a clear format, assuming that the number of servers $m=m(c,r)$ is defined by minimizing the total cost $C(c,m,r)$.