**Remark 4.**


$$\begin{aligned} \sum\_{k=1}^{c} \sum\_{j=0}^{a-1} P^+(I(k), j) + \sum\_{j=0}^{\infty} P^+(B, j) &= \frac{\sum\_{k=1}^{c-1} \sum\_{j=0}^{a-1} P(I(k), j) + \sum\_{j=0}^{\infty} P(B, j)}{1 - \sum\_{i=0}^{a-1} P(I(c), i)} \\ &= \frac{1 - \sum\_{i=0}^{a-1} P(I(c), i)}{1 - \sum\_{i=0}^{a-1} P(I(c), i)} = 1, \end{aligned}$$

*as it should be.*

#### **7. Numerical Results**

In this section, we present some numerical results for various inter-arrival time distributions such as *η*-phase Erlang (E*η*), deterministic (D), and uniform (U). All the examples we considered have the same mean value of the inter-arrival time *E*(*A*) = 1/*λ*. The root equation (see Equation (14)), probability density functions (p.d.f.) of inter-arrival time *A*, and p.d.f. of a random period time *R* for these three distributions are summarized in Table 5.

**Table 5.** Root Equations, p.d.f.s of *<sup>A</sup>*(*t*), *<sup>R</sup>*(*t*), and mean value of of *A*(*t*) for <sup>E</sup>*η*/M*<sup>a</sup>*, *<sup>b</sup>*/*<sup>c</sup>*, D/M*<sup>a</sup>*, *<sup>b</sup>*/*c* and U/M*<sup>a</sup>*, *<sup>b</sup>*/*<sup>c</sup>*.


Besides the calculations for the queue-length probabilities at the pre-arrival, random, and post-departure epochs for both idle and busy systems, we also considered the performance measures, such as the mean (denoted as LQe) and the standard deviations (denoted as SDLQe) of the queue length; the mean (denoted as *Ee*[*I*(*k*)]) and variance (denoted as *Vare*[*I*(*k*)] ) of the idle servers. The symbol "e" denotes the epoch state, which can be pre-arrival (*e* = "−"), random (*e* = " "), or post-departure (*e* = "+"). We define *PBe* = ∑∞*<sup>n</sup>*=<sup>0</sup> *Pe*(*<sup>B</sup>*, *n*) as the probability that an arriving customer sees the system busy at *e* epoch, and *PIe* = ∑*<sup>a</sup>*−<sup>1</sup> *<sup>n</sup>*=0 ∑*ck*=<sup>1</sup> *Pe*(*I*(*k*), *n*) is the probability that the system is idle at *e* epoch. The probabilities of the queue length at three different epochs are presented in closed form. Since most of these probabilities are irrational, for computational purposes, we need to set the precision *ε*. Throughout all computations in

the following examples, we use *ε* = 10−<sup>20</sup> as the precision. Due to the rounding error, the sum of the probabilities may not be one.

The results of the E6/M5,10/5 queue with traffic intensities *ρ* = 0.1, 0.5, 0.9 for both busy and idle servers at pre-arrival epoch are presented in Tables 6 and 7, respectively. When we set the number of servers to 1, our results match with those obtained for E6/M5,10/1 by Chaudhry et al. [5].

We considered three systems E6/M*<sup>a</sup>*,10/5, D/M*<sup>a</sup>*,10/5, and U/M*<sup>a</sup>*,10/5 (*<sup>t</sup>*1 = 0.875/*λ*, *t*2 = 1.125/*λ*, *ϕ* = 0.25/*λ*). All three systems have the same mean value of inter-arrival time *E*(*A*) = 1/*λ*. In Table 8, we present the performance measures for these three systems for idle servers at three different epochs with varied *a* = 1, 4, 7 and *ρ* = 0.1, 0.5, 0.9. In Figure 1, we compare the performance of D/M4,10/5 for busy servers at pre-arrival epochs with *ρ* = 0.1, 0.3, 0.5, 0.7 and 0.9. In Figure 2, we compare the performance of U/M*<sup>a</sup>*,10/5 for busy servers at pre-arrival epochs with *a* = 1, 4, 7.

**Table 6.** Distribution of queue lengths at pre-arrival epochs for the busy system E6/M5,10/5, *ρ* = 0.1, 0.5, 0.9,*ε* = 10−20.


*ρ*


**Table 7.** Distribution of queue lengths at the pre-arrival epochs for the idle system E6/M5,10/5, = 0.1, 0.5, 0.9,*ε* = 10−20.

**Figure 1.** Comparison of performance measures of D/M4,10/5 for busy servers, *ρ* = 0.1, 0.3, 0.5, 0.7, 0.9, *ε* = 10−20.


**Table 8.** Comparison of performance measures of E6/M*a*,10/5, D/M*a*,10/5, and U/M*a*,10/5 for idle servers, *a* = 1, 4, 7, *ρ* = 0.1, 0.5, 0.9, *ε* = 10−20.

**Figure 2.** Comparison of performance measures of U/M*<sup>a</sup>*,10/5 for busy servers, a = 1, 4, 7, *ρ* = 0.5, *ε* = 10−20.
