Event Generation

The important moments in a queuing system are the moments when a call arrives, when a call enters a server, and when a call leaves the system. We call these moments as events. Generating events one needs two types of random number generators: one generating the time until the next call and another for generating the service time.

Denote by $F_a(t)=P_a\{\tau < t\}$ the distribution function defining the probability that a random time $\tau$ will be less then $t$. Here $a$ means the expected value of $\tau$. Denote by $\xi \in [0,1]$ the random variable uniformly distributed between zero and one. Then

$$\tau=F^{-1}(\xi)$$ (80)

In exponential cases

$$F_a(t)=1-e^{-1/a t}$$ (81)

$$\tau=-a ln (1-\xi).$$ (82)

Generating next arrivals of l $a=1/\lambda_s$.
Generating service times $a=1/(m_s \mu_s)$.
Moments when a call enters a server and leaves the system moments depends on the specific structure of the system and are defined by the model. The is designed trying to represent the actual operations as realistically as possible. Outline an algorithm of modeling and optimization of call centers

• denote by $t_n$ the "arriving" time of the $n$th call, then $t_{n+1}=t_n+\xi_n(\lambda),\ \xi(\lambda)= - 1/\lambda\ log(1-\eta)$ where $\eta$ is a random number uniform in $[0,1]$ and $\lambda$ is the average number of calls in a time unit, the first call moment $t_1=0$ is the start of MCS
• 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 $1 \le m_n \le m$ 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 $m$ servers and $r$ waiting places
  

  $$I_{n+1} = \cases {I_n-1,$$ (83)

  
  
  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 $\tau_n$
  

  $$m_n=\min (I_l,m),\ t_l \le \tau_n \le t_{l+1}$$ (84)

  
  

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

  $$L_{n+1} = \cases {L_n+1,$$ (85)

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

  $$T_{n+1} =T_n+\delta_{n},$$ (86)

  $$\delta_{n}=\tau_{n}-t_{n}$$ (87)

  
  

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

  $$q_{n+1}^k = \cases {\min(q_n^k+1,m+r),$$ (88)

  
  

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

  $$Q(t^k)=1/N\ q_{N}^k$$ (89)

  
  

• The optimization of number $m$ is performed by simple comparison of different $m$ within some reasonable bounds $m_{min} \le m \le m_{max}$
  

  $$m(c,r)=arg \min_{m_{min} \le m \le m_{max}} C(c,m,r)$$ (90)

  
  
  .
• the results are compared with the corresponding steady-state explicit solutions including the average waiting time, the average of lost calls and the waiting time distribution functions