A Survey on Queueing Systems with Mathematical Models and Applications

Read this paper on waiting line analysis and queues. It provides a good survey of the theory and uses of this type of analysis. Pay particular attention to sections 1 through 3. How might these models be used to balance firm costs with different levels of customer satisfaction?

4. Formulation of Queueing Models

4.1. Birth-Death Process

A Birth-Death process is a Markov process in which states are numbered by an integer and transitions are only permitted between two neighbouring states. Births are the cases when state variables are increased by one and deaths are the cases when state variables are decreased by one. When birth occurs, the state N moves to state N + 1 and when the death occurs, state N changes to state N - 1.

Figure 2 shows the simple birth-death process with which we can establish the balance equations as follows:

State \quad Rate in = Rate out

\begin{array}{ll}0: & \mu_{1} P_{1}=\lambda_{0} P_{0} \\ 1: & \lambda_{0} P_{0}+\mu_{2} P_{2}=\left(\lambda_{1}+\mu_{1}\right) P_{1}\end{array}

2: \quad \lambda_{1} P_{1}+\mu_{3} P_{3}=\left(\lambda_{2}+\mu_{2}\right) P_{2}

\begin{array}{lll}\cdots & \cdots & \cdots\end{array}

{ }^{N-1}{ }_{-1}={\lambda}_{N-2}{ }^{P}{ }_{N-2}+\mu_{N}{ }^{P}{ }_{N}=\left(\lambda_{N-1}+\mu_{N-1}\right) P_{N-1}

\mathrm{N}: \quad \lambda_{\mathrm{N}-1} \mathrm{P}_{\mathrm{N}-1}+\mu_{\mathrm{N}+1} \mathrm{P}_{\mathrm{N}+1}=\left(\lambda_{\mathrm{N}}+\mu_{\mathrm{N}}\right) \mathrm{P}_{\mathrm{N}}

All the above balanced equations can be expressed in terms of \mathrm{P}_{0} as follows:

0: \quad \mathrm{P}_{1}=\frac{\lambda_{0}}{\mu_{1}} \mathrm{P}_{0}

1: \quad P_{2}=\frac{\lambda_{1}}{\mu_{2}} P_{1}+\frac{\left(\mu_{1} P_{1}-\lambda_{0} P_{0}\right)}{\mu_{2}}

\begin{aligned}&=\frac{\lambda_{1}}{\mu_{2}} P_{1}+\frac{\left(\mu_{1} P_{1}-\mu_{1} P_{1}\right)}{\mu_{2}} \\ &=\frac{\lambda_{1}}{\mu_{2}} \frac{\lambda_{0}}{\mu_{1}} P\end{aligned}

Continuing this way

 \begin{aligned} N-1: \qquad P_{N} &=\frac{\lambda_{N-1}}{\mu_{N}} P_{N-1}+\frac{\left(\mu_{N-1} P_{N-1}-\lambda_{N-2} P_{N-2}\right)}{\mu_{N}} \\ &=\frac{{ }{\lambda}_{\mathrm{~N}-1}}{\mu_{N}} P_{N-1}+\frac{\left(\mu_{N-1} P_{N-1}-\mu_{N-1} P_{N-1}\right)}{\mu_{N}} \\ &=\frac{\lambda_{{N}-1}}{\mu_{N}} P_{N-1}\end{aligned}

 \begin{aligned} & N: \quad P_{N+1} =\frac{\lambda_{N}}{\mu_{N-1}} P_{N}+\frac{\left(\mu_{N} P_{N}-\lambda_{N-1} P_{N-1}\right)}{\mu_{N-1}}\\ & \qquad \qquad \quad =\frac{\lambda_{N}}{\mu_{N-1}} P_{N}+\frac{\left(\mu_{N} P_{N}-\mu_{N} P_{N}\right)}{\mu_{N-1}}\\ &  \qquad \qquad \quad =\frac{\lambda_{N}}{\mu_{N-1}} P_{N}\\ &  \qquad \qquad \quad =\frac{\lambda_{N}{N}_{-1} \cdots \lambda_{0}}{\mu_{N-1} m_{k} \cdots \mu_{1}} P_{0}\\ & \text { If } \mathrm{C}_{\mathrm{N}}=\frac{\lambda_{\mathrm{N}-1}{ }^{\lambda} \mathrm{N}-2 \cdots \lambda_{0}}{\mu_{\mathrm{N}}+\mu_{\mathrm{N}-1} \ldots \mu_{1}}, \text { then } \mathrm{P}_{\mathrm{N}}=\mathrm{C}_{\mathrm{N}} \mathrm{P}_{0} \end{aligned}