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Asymptotic upper bounds for an M/M/C/K retrial queue with a guard channel and guard buffer

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Abstract

The paper deals with Markovian multiserver retrial queuing system with exponential abandonments, two types of arrivals: Fresh calls and Handover calls and waiting places in the service area. This model can be used for analysing a cellular mobile network, where the service area is divided into cells. In this paper, the number of customers in the system and in the orbit form a level-dependent quasi-birth-and-death process, whose stationary distribution is expressed in terms of a sequence of rate matrices. First, we derive the Taylor series expansion for nonzero elements of the rate matrices. Then, by the expansion results, we obtain an asymptotic upper bound for the stationary distribution of both the number of busy channels and the number of customers in the orbit. Furthermore, we present some numerical results to examine the performance of the system.

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Appendix A

Appendix A

1.1 A.1 Proof of Theorem 5

Proof

We prove Theorem 5 using mathematical induction. We show that is true for \(k=1\), \(k=\overline{2, K-1}\), \(k=K\) and \(k=0\) respectively.

  • The case \(k=1\):

    According to Theorem 4, for \(i=0, 1, C,\ldots , K-C,\ldots ,K-1\)

    $$\begin{aligned}{} & {} r^{(0, j)}_{i}=o(1), \ r^{(1, j)}_{i}=o(1), \end{aligned}$$
    (A1)
    $$\begin{aligned}{} & {} r^{(0, j)}_{i}\le \frac{\lambda _{2}}{j\theta }, \ r^{(1, j)}_{i}\le \frac{\lambda }{j\theta }. \end{aligned}$$
    (A2)

    Furthermore, it follows from (11) and (17) for \(i=0\) that

    $$\begin{aligned} r^{(0, j)}_{0}= & {} \frac{1}{j\theta }\left( -\lambda r^{(0, j)}_{0}+\mu r^{(0, j)}_{1}\right) , \end{aligned}$$
    (A3)
    $$\begin{aligned} r^{(1, j)}_{0}= & {} \frac{1}{j\theta }\left( -\lambda r^{(1, j)}_{0}+\mu r^{(1, j)}_{1}\right) . \end{aligned}$$
    (A4)

    From (A1), (A3) and (A4), we obtain

    $$\begin{aligned} r^{(0, j)}_{0}=o\left( \frac{1}{j}\right) , \ r^{(1, j)}_{0}=o\left( \frac{1}{j}\right) . \end{aligned}$$
    (A5)

    In addition, it follows from (16) and (21) that

    $$\begin{aligned} \left( \lambda +C\mu +(K-C)\gamma \right) r^{(0, j)}_{K}= & {} \lambda _{1} r^{(0, j)}_{K-1}+\left( j+1\right) \theta \left( r^{(0, j)}_{K-1}r^{(0, j+1)}_{K-1}+r^{(0, j)}_{K}r^{(1, j+1)}_{K-1}\right) ,\\ \left( \lambda +C\mu +(K-C)\gamma \right) r^{(1, j)}_{K}= & {} \lambda _{1} r^{(1, j)}_{K-1}+\lambda +\left( j+1\right) \theta \left( r^{(1, j)}_{K-1}r^{(0, j+1)}_{K-1}+r^{(1, j)}_{K}r^{(1, j+1)}_{K-1}\right) . \end{aligned}$$

    From (A2), we obtain

    $$\begin{aligned} \left( \lambda +C\mu +(K-C)\gamma \right) r^{(0, j)}_{K}\le & {} \lambda _{1} r^{(0, j)}_{K-1}+\lambda _{2}r^{(0, j)}_{K-1}+\lambda r^{(0, j)}_{K},\\ \left( \lambda +C\mu +(K-C)\gamma \right) r^{(1, j)}_{K}\le & {} \lambda _{1} r^{(1, j)}_{K-1}+\lambda _{2}r^{(1, j)}_{K-1}+\lambda r^{(1, j)}_{K}+\lambda . \end{aligned}$$

    Deleting \(\lambda r^{(0, j)}_{K}\) and \(\lambda r^{(1, j)}_{K}\) from both sides yields

    $$\begin{aligned} \left( C\mu +(K-C)\gamma \right) r^{(0, j)}_{K}\le & {} \lambda _{1} r^{(0, j)}_{K-1}+\lambda _{2}r^{(0, j)}_{K-1},\\ \left( C\mu +(K-C)\gamma \right) r^{(1, j)}_{K}\le & {} \lambda _{1} r^{(1, j)}_{K-1}+\lambda _{2}r^{(1, j)}_{K-1}+\lambda . \end{aligned}$$

    From (A1), we obtain

    $$\begin{aligned} r^{(0, j)}_{K}=o(1), \ r^{(1, j)}_{K}=o(1). \end{aligned}$$
    (A6)

    From (11) and (A2) for \(i=1,\ldots , C-2\), we have

    $$\begin{aligned} r^{(0, j)}_{i}= & {} \frac{1}{j\theta }\left( \lambda r^{(0, j)}_{i-1}-(\lambda +i\mu )r^{(0, j)}_{i}+(i+1)\mu r^{(0, j)}_{i+1}+\left( j+1\right) \theta r^{(0, j)}_{K-1}r^{(0, j+1)}_{i-1}\right. \\{} & {} \left. +(j+1)\theta r^{(0, j)}_{K}r^{(1, j+1)}_{i-1}\right) ,\\ r^{(0, j)}_{i}\le & {} \frac{1}{j\theta }\left( \lambda r^{(0, j)}_{i-1}-(\lambda +i\mu )r^{(0, j)}_{i}+(i+1)\mu r^{(0, j)}_{i+1}+\lambda _{2} r^{(0, j)}_{K-1}+\lambda r^{(0, j)}_{K}\right) . \end{aligned}$$

    From (12) and (A2) for \(i=C-1\), we have

    $$\begin{aligned} r^{(0, j)}_{i}= & {} \frac{1}{j\theta }\left( \lambda r^{(0, j)}_{i-1}-(\lambda _{1}+i\mu )r^{(0, j)}_{i}+(i+1)\mu r^{(0, j)}_{i+1}+(j+1)\theta r^{(0, j)}_{K-1}r^{(0, j+1)}_{i-1} \right. \\{} & {} \left. +(j+1)\theta r^{(0, j)}_{K}r^{(1, j+1)}_{i-1}\right) ,\\ r^{(0, j)}_{i}\le & {} \frac{1}{j\theta }\left( \lambda r^{(0, j)}_{i-1}-(\lambda _{1}+i\mu )r^{(0, j)}_{i}+(i+1)\mu r^{(0, j)}_{i+1}+\lambda _{2} r^{(0, j)}_{K-1}+\lambda r^{(0, j)}_{K}\right) . \end{aligned}$$

    From (13) and (A2) for \(i=C\), we have

    $$\begin{aligned} r^{(0, j)}_{i}= & {} \frac{1}{j\theta }\left( \lambda _{1} r^{(0, j)}_{i-1}-(\lambda +C\mu +(i-C)\gamma )r^{(0, j)}_{i}+(C\mu +(i+1-C)\gamma ) r^{(0, j)}_{i+1}\right. \\{} & {} \left. +(j+1)\theta r^{(0, j)}_{K-1}r^{(0, j+1)}_{i-1}+(j+1)\theta r^{(0, j)}_{K}r^{(1, j+1)}_{i-1}\right) ,\\ r^{(0, j)}_{i}\le & {} \frac{1}{j\theta }\left( \lambda _{1} r^{(0, j)}_{i-1}-(\lambda +C\mu +(i-C)\gamma )r^{(0, j)}_{i}+(C\mu +(i+1-C)\gamma ) r^{(0, j)}_{i+1}\right. \\{} & {} \left. +\lambda _{2} r^{(0, j)}_{K-1}+\lambda r^{(0, j)}_{K}\right) . \end{aligned}$$

    From (14) and (A2) for \(i=C+1,\ldots , K-2\), we have

    $$\begin{aligned} r^{(0, j)}_{i}= & {} \frac{1}{j\theta }\left( \lambda r^{(0, j)}_{i-1}-(\lambda +C\mu +(i-C)\gamma )r^{(0, j)}_{i}+(C\mu +(i+1-C)\gamma ) r^{(0, j)}_{i+1}\right. \\{} & {} \left. +(j+1)\theta r^{(0, j)}_{K-1}r^{(0, j+1)}_{i-1}+(j+1)\theta r^{(0, j)}_{K}r^{(1, j+1)}_{i-1}\right) ,\\ r^{(0, j)}_{i}\le & {} \frac{1}{j\theta }\left( \lambda r^{(0, j)}_{i-1}-(\lambda +C\mu +(i-C)\gamma )r^{(0, j)}_{i}+(C\mu +(i+1-C)\gamma )r^{(0, j)}_{i+1} \right. \\{} & {} \left. +\lambda _{2} r^{(0, j)}_{K-1}+\lambda r^{(0, j)}_{K}\right) . \end{aligned}$$

    It follows from (A1) and (A6) that

    $$\begin{aligned} r^{(0, j)}_{i}=o\left( \frac{1}{j}\right) ,\ i=1,\ldots , C,\ldots ,K-2. \end{aligned}$$
    (A7)

    From Theorem 4, (A5) and (A7), we obtain

    $$\begin{aligned} r^{(0, j)}_{K-1}=\frac{\lambda _{2}}{j\theta }+o\left( \frac{1}{j}\right) . \end{aligned}$$
    (A8)

    Thus, we obtain \(r^{(0, j)}_{i}=o(\frac{1}{j})\), \(i= 0, 1,\ldots , K-2\), \(r^{(0, j)}_{K-1}=\frac{h^{(0, 1)}_{0}}{j}+o(\frac{1}{j})\). Arranging (17), (18), (19) and (20) respectively, yields

    $$\begin{aligned} j\theta r^{(1, j)}_{i}= & {} -(\lambda +i\mu ) r^{(1, j)}_{i}+\lambda r^{(1, j)}_{i-1}+(i+1)\mu r^{(1, j)}_{i+1}+\widetilde{r}^{(1, j)}_{i},\nonumber \\{} & {} \text {for} \ 0\le i\le C-2; \end{aligned}$$
    (A9)
    $$\begin{aligned} j\theta r^{(1, j)}_{i}= & {} -(\lambda _{1}+i\mu ) r^{(1, j)}_{i}+\lambda r^{(1, j)}_{i-1}+(C\mu +(i+1-C)\gamma )r^{(1, j)}_{i+1}\nonumber \\{} & {} +\widetilde{r}^{(1, j)}_{i}, \ \text {for} \ i=C-1; \end{aligned}$$
    (A10)
    $$\begin{aligned} j\theta r^{(1, j)}_{i}= & {} -(\lambda +C\mu +(i-C)\gamma ) r^{(1, j)}_{i}+(C\mu +(i+1-C)\gamma ) r^{(1, j)}_{i+1} \nonumber \\{} & {} +\lambda _{1} r^{(1, j)}_{i-1}+\widetilde{r}^{(1, j)}_{i},\ \text {for} \ i=C; \end{aligned}$$
    (A11)
    $$\begin{aligned} j\theta r^{(1, j)}_{i}= & {} -(\lambda +C\mu +(i-C)\gamma ) r^{(1, j)}_{i}+(C\mu +(i+1-C)\gamma ) r^{(1, j)}_{i+1} \nonumber \\{} & {} +\lambda r^{(1, j)}_{i-1}+\widetilde{r}^{(1, j)}_{i},\ \text {for} \ C+1\le i\le K-1. \end{aligned}$$
    (A12)

    It follows from (A1), (A5) and (A6) that

    $$\begin{aligned} r^{(1, j)}_{0}= & {} o\left( \frac{1}{j}\right) ,\ r^{(1, j)}_{1}=o(1),\ r^{(1, j)}_{2}=o(1),\\ r^{(0, j+1)}_{0}= & {} o\left( \frac{1}{j+1}\right) ,\ r^{(1, j+1)}_{0}=o\left( \frac{1}{j+1}\right) ,\\ r^{(1, j)}_{K-1}= & {} o(1),\ r^{(1, j)}_{K}=o(1). \end{aligned}$$

    Substituting the above formulae into (A9)-(A12) with \(i=1\) yields \(r^{(1, j)}_{1}=o(\frac{1}{j})\). We assume that Theorem 5 is true for \(i=n-1\), i.e., \(r^{(1, j)}_{n-1}=o(\frac{1}{j})\).

    From the preceding assumption, (A1), (A5) and (A6), we have

    $$\begin{aligned} r^{(1, j)}_{n-1}= & {} o\left( \frac{1}{j}\right) ,\ r^{(1, j)}_{n}=o(1),\ r^{(1, j)}_{n+1}=o(1),\\ r^{(0, j+1)}_{n-1}= & {} o\left( \frac{1}{j+1}\right) ,\ r^{(1, j+1)}_{n-1}=o\left( \frac{1}{j+1}\right) ,\\ r^{(1, j)}_{K-1}= & {} o(1),\ r^{(1, j)}_{K}=o(1). \end{aligned}$$

    Substituting these formulae into (A9)-(A12) with \(i=n\), we obtain \(r^{(1, j)}_{n}=o(\frac{1}{j})\). Using mathematical induction, we have \(r^{(1, j)}_{i}=o(\frac{1}{j})\) for \(i=1, 2,\ldots , K-2\), which together with Theorem 4 and (A5) yields

    $$\begin{aligned} r^{(1, j)}_{K-1}=\frac{\lambda }{j\theta }+o\left( \frac{1}{j}\right) . \end{aligned}$$
    (A13)

    Thus, we obtain \(r^{(1, j)}_{i}=o(\frac{1}{j})\), \(i= 0, 1,\ldots , K-2\), \(r^{(0, j)}_{K-1}=\frac{h^{(1, 1)}_{0}}{j}+o(\frac{1}{j})\).

  • The case \(k=2, 3,\ldots , K-1\):

    It should be noted that the derivations for \(r^{(0, j)}_{K-k}\) and \(r^{(1, j)}_{K-k}\) are the same. Thus, we show for \(r^{(0, j)}_{K-k}\) only. For \(k=1, 2,\ldots , n\), we assume that

    $$\begin{aligned}{} & {} r^{(0, j)}_{K-k}=h^{(0, j)}_{0}\frac{1}{j^{k}}+o\left( \frac{1}{j^{k}}\right) , \ j\in N; \end{aligned}$$
    (A14)
    $$\begin{aligned}{} & {} r^{(1, j)}_{K-k}=h^{(1, j)}_{0}\frac{1}{j^{k}}+o\left( \frac{1}{j^{k}}\right) , \ j\in N; \end{aligned}$$
    (A15)
    $$\begin{aligned}{} & {} r^{(0, j)}_{i}=o\left( \frac{1}{j^{k}}\right) , \ i=0, 1,\ldots ,K-k-1; \end{aligned}$$
    (A16)
    $$\begin{aligned}{} & {} r^{(1, j)}_{i}=o\left( \frac{1}{j^{k}}\right) , \ i=0, 1,\ldots ,K-k-1. \end{aligned}$$
    (A17)

    We prove that the same expression is obtainable for the case \(k=n+1\). Indeed, it follows from (A3), (A4), (A16) and (A17) that

    $$\begin{aligned} r^{(0, j)}_{0}=o\left( \frac{1}{j^{n+1}}\right) , \ r^{(1, j)}_{0}=o\left( \frac{1}{j^{n+1}}\right) . \end{aligned}$$

    For \(i=1, 2,\ldots , K-n-2\), assuming that \(r^{(0, j)}_{i-1}=o(\frac{1}{j^{n+1}})\) and \(r^{(1, j)}_{i-1}=o(\frac{1}{j^{n+1}})\), we prove that \(r^{(0, j)}_{i}=o(\frac{1}{j^{n+1}})\) and \(r^{(1, j)}_{i}=o(\frac{1}{j^{n+1}})\).

    Indeed, arranging (11) and (21) yields

    $$\begin{aligned} r^{(0, j)}_{i}= & {} \frac{1}{j\theta }\left( \lambda r^{(0, j)}_{i-1}-(\lambda +i\mu ) r^{(0, j)}_{i}+(i+1)\mu r^{(0, j)}_{i+1}+\widetilde{r}^{(0, j)}_{i}\right) ,\nonumber \\{} & {} \text {for} \ 0\le i\le C-2; \end{aligned}$$
    (A18)
    $$\begin{aligned} r^{(0, j)}_{i}= & {} \frac{1}{j\theta }(\lambda r^{(0, j)}_{i-1}-(\lambda _{1}+i\mu )r^{(0, j)}_{i}+(i+1)\mu r^{(0, j)}_{i+1}\nonumber \\{} & {} +\widetilde{r}^{(0, j)}_{i}), \ \text {for} \ i=C-1; \end{aligned}$$
    (A19)
    $$\begin{aligned} r^{(0, j)}_{i}= & {} \frac{1}{j\theta }(\lambda _{1} r^{(0, j)}_{i-1}-(\lambda +C\mu +(i-C)\gamma ) r^{(0, j)}_{i} \nonumber \\{} & {} +(C\mu +(i+1-C)\gamma )r^{(0, j)}_{i+1}+\widetilde{r}^{(0, j)}_{i}),\ \text {for} \ i=C; \end{aligned}$$
    (A20)
    $$\begin{aligned} r^{(0, j)}_{i}= & {} \frac{1}{j\theta }(\lambda r^{(0, j)}_{i-1}-(\lambda +C\mu +(i-C)\gamma )r^{(0, j)}_{i} \nonumber \\{} & {} +(C\mu +(i+1-C)\gamma )r^{(0, j)}_{i+1}+\widetilde{r}^{(0, j)}_{i}), \ \text {for} \ C+1\le i\le K-2; \end{aligned}$$
    (A21)
    $$\begin{aligned} r^{(1, j)}_{i}= & {} \frac{1}{j\theta }\left( \lambda r^{(1, j)}_{i-1}-(\lambda +i\mu ) r^{(1, j)}_{i}+(i+1)\mu r^{(1, j)}_{i+1}+\widetilde{r}^{(1, j)}_{i}\right) , \nonumber \\{} & {} \text {for} \ 0\le i\le C-2; \end{aligned}$$
    (A22)
    $$\begin{aligned} r^{(1, j)}_{i}= & {} \frac{1}{j\theta }(\lambda r^{(1, j)}_{i-1}-(\lambda _{1}+i\mu ) r^{(1, j)}_{i}+(i+1)\mu r^{(1, j)}_{i+1} \nonumber \\{} & {} +\widetilde{r}^{(1, j)}_{i}), \ \text {for} \ i=C-1; \end{aligned}$$
    (A23)
    $$\begin{aligned} r^{(1, j)}_{i}= & {} \frac{1}{j\theta }(\lambda _{1} r^{(1, j)}_{i-1}-(\lambda +C\mu +(i-C)\gamma ) r^{(1, j)}_{i} \nonumber \\{} & {} +(C\mu +(i+1-C)\gamma )r^{(1, j)}_{i+1}+\widetilde{r}^{(1, j)}_{i}),\ \text {for} \ i=C; \end{aligned}$$
    (A24)
    $$\begin{aligned} r^{(1, j)}_{i}= & {} \frac{1}{j\theta }(\lambda r^{(1, j)}_{i-1}-(\lambda +C\mu +(i-C)\gamma ) r^{(1, j)}_{i} \nonumber \\{} & {} +(C\mu +(i+1-C)\gamma ) r^{(1, j)}_{i+1}+\widetilde{r}^{(1, j)}_{i}),\ \text {for} \ C+1\le i\le K-1. \qquad \end{aligned}$$
    (A25)

    Applying the preceding assumption, (A6), (A8), (A16) and (A17) to (A18)-(A21) yields

    $$\begin{aligned} r^{(0, j)}_{i}=o\left( \frac{1}{j^{n+1}}\right) , \ i=1, 2, \ldots , K-n-2. \end{aligned}$$
    (A26)

    Similarly, substituting the preceding assumption, (A6), (A13), (A16) and (A17) to (A22)-(A25), we obtain

    $$\begin{aligned} r^{(1, j)}_{i}=o\left( \frac{1}{j^{n+1}}\right) , \ i=1, 2, \ldots , K-n-2. \end{aligned}$$
    (A27)

    It follows from (A14), (A15), (A18)-(A25), (A26) and (A27) that

    $$\begin{aligned}{} & {} r^{(0, j)}_{K-n-1}=h^{(0, j+1)}_{0}\frac{1}{j^{n+1}}+o\left( \frac{1}{j^{n+1}}\right) , \ j\in N;\\{} & {} r^{(1, j)}_{K-n-1}=h^{(1, j+1)}_{0}\frac{1}{j^{n+1}}+o\left( \frac{1}{j^{n+1}}\right) , \ j\in N. \end{aligned}$$

    Thus, we have proven the case \(k=n+1\). As a result, we have proven for \(k=2,\ldots , K-1\).

  • The case \(k=K\):

    Substituting (A14), (A15), (A16), (A17) with \(k=K-1\) into (A3) and (A4), we obtain

    $$\begin{aligned} r^{(0, j)}_{0}= & {} h^{(0, K)}_{0}\frac{1}{j^{K}}+o\left( \frac{1}{j^{K}}\right) , \ j\in N;\\ r^{(1, j)}_{0}= & {} h^{(1, K)}_{0}\frac{1}{j^{K}}+o\left( \frac{1}{j^{K}}\right) , \ j\in N. \end{aligned}$$
  • The case \(k=0\):

    Arranging (16) and (21), we obtain

    $$\begin{aligned}{} & {} \left( \lambda +C\mu +(K-C)\gamma \right) r^{(0, j)}_{K}=\lambda _{1} r^{(0, j)}_{K-1}+\left( j+1\right) \theta r^{(0, j)}_{K-1}r^{(0, j+1)}_{K-1} \nonumber \\{} & {} \quad +(j+1)\theta r^{(0, j)}_{K}r^{(1, j+1)}_{K-1}), \end{aligned}$$
    (A28)
    $$\begin{aligned}{} & {} \left( \lambda +C\mu +(K-C)\gamma \right) r^{(1, j)}_{K}=\lambda _{1} r^{(1, j)}_{K-1}+\lambda +\left( j+1\right) \theta r^{(1, j)}_{K-1}r^{(0, j+1)}_{K-1} \nonumber \\{} & {} \quad +(j+1)\theta r^{(1, j)}_{K}r^{(1, j+1)}_{K-1}). \end{aligned}$$
    (A29)

    From (A14) and (A15) with \(k=1\), we obtain

    $$\begin{aligned} \left( j+1\right) r^{(0, j+1)}_{K-1}= & {} \frac{\lambda _{2}}{\theta }+o(1), \ j\in N;\\ \left( j+1\right) r^{(1, j+1)}_{K-1}= & {} \frac{\lambda }{\theta }+o(1), \ j\in N. \end{aligned}$$

    Substituting the above two formulae into (A28) and (A29) yields

    $$\begin{aligned}{} & {} \left( \lambda +C\mu +(K-C)\gamma \right) r^{(0, j)}_{K}=\lambda _{1} r^{(0, j)}_{K-1}+\lambda _{2} r^{(0, j)}_{K-1}+\lambda r^{(0, j)}_{K}+o(1),\\{} & {} \quad \left( \lambda +C\mu +(K-C)\gamma \right) r^{(1, j)}_{K}=\lambda _{1} r^{(1, j)}_{K-1}+\lambda +\lambda _{2} r^{(1, j)}_{K-1}+\lambda r^{(1, j)}_{K}+o(1). \end{aligned}$$

    Deleting \(\lambda r^{(0, j)}_{K}\) and \(\lambda r^{(1, j)}_{K}\) from both sides yields

    $$\begin{aligned} \left( C\mu +(K-C)\gamma \right) r^{(0, j)}_{K}= & {} \lambda _{1} r^{(0, j)}_{K-1}+\lambda _{2} r^{(0, j)}_{K-1}+o(1),\\ \left( C\mu +(K-C)\gamma \right) r^{(1, j)}_{K}= & {} \lambda _{1} r^{(1, j)}_{K-1}+\lambda +\lambda _{2} r^{(1, j)}_{K-1}+o(1). \end{aligned}$$

    From these two formulae and the result for \(k=1\), we obtain

    $$\begin{aligned} r^{(0, j)}_{K}= & {} h^{(0, 0)}_{0}+o(1), \ j\in N;\\ r^{(1, j)}_{K}= & {} h^{(1, 0)}_{0}+o(1), \ j\in N. \end{aligned}$$

    where

    $$\begin{aligned} h^{(0, 0)}_{0}=0, \ h^{(1, 0)}_{0}=\frac{\lambda }{C\mu +(K-C)\gamma }. \end{aligned}$$

\(\square \)

1.2 A.2 Proof of Theorem 6

Proof

We prove Theorem 6 for \(k= 0, 1,\ldots , K\) using mathematical induction.

  • The case  \(k=1\): From Theorem 4, we have

    $$\begin{aligned} r^{(0, j)}_{K-1}= & {} \frac{\lambda _{2}}{j\theta }-r^{(0, j)}_{K-2}-\sum ^{K}_{k=3}r^{(0, j)}_{K-k},\\ r^{(1, j)}_{K-1}= & {} \frac{\lambda }{j\theta }-r^{(1, j)}_{K-2}-\sum ^{K}_{k=3}r^{(1, j)}_{K-k}. \end{aligned}$$

    Furthermore, from Theorem 5, we have

    $$\begin{aligned} r^{(0, j)}_{K-2}= & {} h^{(0, 2)}_{0}\frac{1}{j^{2}}+o\left( \frac{1}{j^{2}}\right) =O\left( \frac{1}{j^{2}}\right) ,\\ r^{(1, j)}_{K-2}= & {} h^{(1, 2)}_{0}\frac{1}{j^{2}}+o\left( \frac{1}{j^{2}}\right) =O\left( \frac{1}{j^{2}}\right) ,\\ \sum ^{K}_{k=3}r^{(0, j)}_{K-k}= & {} O\left( \frac{1}{j^{2}}\right) ,\\ \sum ^{K}_{k=3}r^{(1, j)}_{K-k}= & {} O\left( \frac{1}{j^{2}}\right) .\\ \end{aligned}$$

    Thus, we obtain

    $$\begin{aligned} r^{(0, j)}_{K-1}= & {} h^{(0, 1)}_{0}\frac{1}{j}+O\left( \frac{1}{j^{2}}\right) ,\\ r^{(1, j)}_{K-1}= & {} h^{(1, 1)}_{0}\frac{1}{j}+O\left( \frac{1}{j^{2}}\right) . \end{aligned}$$
  • The case \(k=2, 3,\ldots , K-1\):

    We assume (25) and (26) are true for \(r^{(0, j)}_{K-n}\) with \(n=1, 2,\ldots , k-1\), we prove that they are also true for \(n=k\).

    Arranging (11)-(14) and (17)-(20) with \(i=K-k\) yields

    $$\begin{aligned} r^{(0, j)}_{K-k}= & {} \frac{\lambda r^{(0, j)}_{K-k-1}-(\lambda +(K-k)\mu ) r^{(0, j)}_{K-k}+(K-k+1)\mu r^{(0, j)}_{K-k+1}+\widetilde{r}^{(0, j)}_{K-k}}{j\theta },\nonumber \\{} & {} \text {for} \ 1\le K-k\le C-2; \end{aligned}$$
    (A30)
    $$\begin{aligned} r^{(0, j)}_{K-k}= & {} \frac{\lambda r^{(0, j)}_{K-k-1}-(\lambda _{1}+(K-k)\mu ) r^{(0, j)}_{K-k}+(K-k+1)\mu r^{(0, j)}_{K-k+1}+\widetilde{r}^{(0, j)}_{K-k}}{j\theta },\nonumber \\{} & {} \text {for} \ K-k=C-1; \end{aligned}$$
    (A31)
    $$\begin{aligned} r^{(0, j)}_{K-k}= & {} \frac{\lambda _{1} r^{(0, j)}_{K-k-1}-(\lambda +C\mu +(K-k-C)\gamma ) r^{(0, j)}_{K-k}+(C\mu +(K-k+1-C)\gamma )r^{(0, j)}_{K-k+1}}{j\theta }\nonumber \\{} & {} \frac{+\widetilde{r}^{(0, j)}_{K-k}}{j\theta }, \ \text {for} \ K-k=C; \end{aligned}$$
    (A32)
    $$\begin{aligned} r^{(0, j)}_{K-k}= & {} \frac{\lambda r^{(0, j)}_{K-k-1}-(\lambda +C\mu +(K-k-C)\gamma ) r^{(0, j)}_{K-k}+(C\mu +(K-k+1-C)\gamma )r^{(0, j)}_{K-k+1}}{j\theta }\nonumber \\{} & {} \frac{+\widetilde{r}^{(0, j)}_{K-k}}{j\theta }, \ \text {for} \ C+1\le K-k\le K-2; \end{aligned}$$
    (A33)
    $$\begin{aligned} r^{(1, j)}_{K-k}= & {} \frac{\lambda r^{(1, j)}_{K-k-1}-(\lambda +(K-k)\mu ) r^{(1, j)}_{K-k}+(K-k+1)\mu r^{(1, j)}_{K-k+1}+\widetilde{r}^{(1, j)}_{K-k}}{j\theta },\nonumber \\{} & {} \text {for} \ 1\le K-k\le C-2; \end{aligned}$$
    (A34)
    $$\begin{aligned} r^{(1, j)}_{K-k}= & {} \frac{\lambda r^{(1, j)}_{K-k-1}-(\lambda _{1}+(K-k)\mu ) r^{(1, j)}_{K-k}+(C\mu +(K-k+1-C)\gamma ) r^{(1, j)}_{K-k+1}+\widetilde{r}^{(1, j)}_{K-k}}{j\theta }, \nonumber \\{} & {} \text {for} \ K-k=C-1; \end{aligned}$$
    (A35)
    $$\begin{aligned} r^{(1, j)}_{K-k}= & {} \frac{\lambda _{1} r^{(1, j)}_{K-k-1}-(\lambda +C\mu +(K-k-C)\gamma ) r^{(1, j)}_{K-k}+(C\mu +(K-k+1-C)\gamma )r^{(1, j)}_{K-k+1}}{j\theta } \nonumber \\{} & {} \frac{+\widetilde{r}^{(1, j)}_{K-k}}{j\theta }, \ \text {for} \ K-k=C; \end{aligned}$$
    (A36)
    $$\begin{aligned} r^{(1, j)}_{K-k}= & {} \frac{\lambda r^{(1, j)}_{K-k-1}-(\lambda +C\mu +(K-k-C)\gamma ) r^{(1, j)}_{K-k}+(C\mu +(K-k+1-C)\gamma ) r^{(1, j)}_{K-k+1}}{j\theta } \nonumber \\{} & {} \frac{+\widetilde{r}^{(1, j)}_{K-k}}{j\theta }, \ \text {for} \ C+1\le K-k\le K-2. \end{aligned}$$
    (A37)

    Applying the assumption of mathematical induction, Theorem 5 and (25) with \(k=1\), we obtain

    $$\begin{aligned} r^{(0, j)}_{K-k-1}= & {} h^{(0, k+1)}_{0}\frac{1}{j^{k+1}}+o\left( \frac{1}{j^{k+1}}\right) ,\\ r^{(0, j)}_{K-k}= & {} h^{(0, k)}_{0}\frac{1}{j^{k}}+o\left( \frac{1}{j^{k}}\right) ,\\ r^{(0, j)}_{K-k+1}= & {} h^{(0, k-1)}_{0}\frac{1}{j^{k-1}}+O\left( \frac{1}{j^{k}}\right) ,\\ (j+1)r^{(0, j+1)}_{K-k-1}= & {} h^{(0, k+1)}_{0}\frac{1}{(j+1)^{k}}+o\left( \frac{1}{j^{k}}\right) \\= & {} h^{(0, k+1)}_{0}\frac{1}{j^{k}}+o\left( \frac{1}{j^{k}}\right) ,\\ (j+1)r^{(1, j+1)}_{K-k-1}= & {} h^{(1, k+1)}_{0}\frac{1}{(j+1)^{k}}+o\left( \frac{1}{j^{k}}\right) \\= & {} h^{(1, k+1)}_{0}\frac{1}{j^{k}}+o\left( \frac{1}{j^{k}}\right) ,\\ r^{(0, j)}_{K-1}= & {} h^{(0, 1)}_{0}\frac{1}{j}+O(\frac{1}{j^{2}}),\\ r^{(0, j)}_{K}= & {} 0+o(1). \end{aligned}$$

    Thus, substituting the above formulae to (A30)-(A33) yields

    $$\begin{aligned} r^{(0, j)}_{K-k}= & {} \frac{(K-k+1)\mu }{j\theta }h^{(0, k-1)}_{0}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) =h^{(0, k)}_{0}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) , \\{} & {} for \ K-C+2\le k\le K-1;\\ r^{(0, j)}_{K-k}= & {} \frac{C\mu +(K-k+1-C)\gamma }{\theta }h^{(0, k-1)}_{0}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) =h^{(0, k)}_{0}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) , \\{} & {} for \ k=K-C+1;\\ r^{(0, j)}_{K-k}= & {} \frac{C\mu +(K-k+1-C)\gamma }{\theta }h^{(0, k-1)}_{0}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) =h^{(0, k)}_{0}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) , \\{} & {} for \ k=K-C;\\ r^{(0, j)}_{K-k}= & {} \frac{C\mu +(K-k+1-C)\gamma }{\theta }h^{(0, k-1)}_{0}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) =h^{(0, k)}_{0}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) , \\{} & {} for \ 2\le k\le K-C-1. \end{aligned}$$

    where

    $$\begin{aligned} h^{(0, k)}_{0}=\frac{\lambda _{2}}{\theta }\prod ^{k-1}_{i=1}\frac{C\mu +(K-i-C)\gamma }{\theta }, \ for \ 2\le k\le K-C+1; \end{aligned}$$

    and

    $$\begin{aligned} h^{(0, k)}_{0}=\frac{\lambda _{2}}{\theta }\prod ^{K-C}_{i=1}\frac{C\mu +(K-i-C)\gamma }{\theta }\prod ^{k-1}_{i=K-C+1}\frac{(K-i)\mu }{\theta }, \ for \ K-C+2\le k\le K-1. \end{aligned}$$

    Similarly, it follows from the assumption of mathematical induction, Theorems 5 and 6 with \(k=1\) that

    $$\begin{aligned} r^{(1, j)}_{K-k-1}= & {} h^{(1, k+1)}_{0}\frac{1}{j^{k+1}}+o\left( \frac{1}{j^{k+1}}\right) ,\\ r^{(1, j)}_{K-k}= & {} h^{(1, k)}_{0}\frac{1}{j^{k}}+o\left( \frac{1}{j^{k}}\right) ,\\ r^{(1, j)}_{K-k+1}= & {} h^{(1, k-1)}_{0}\frac{1}{j^{k-1}}+O\left( \frac{1}{j^{k}}\right) ,\\ (j+1)r^{(0, j+1)}_{K-k-1}= & {} h^{(1, k+1)}_{0}\frac{1}{(j+1)^{k}}+o\left( \frac{1}{j^{k}}\right) \\= & {} h^{(1, k+1)}_{0}\frac{1}{j^{k}}+o\left( \frac{1}{j^{k}}\right) ,\\ r^{(1, j)}_{K-1}= & {} h^{(1, 1)}_{0}\frac{1}{j}+O(\frac{1}{j^{2}}),\\ r^{(1, j)}_{K}= & {} h^{(1, 0)}_{0}+o(1). \end{aligned}$$

    Thus, substituting these formulae into (A34)–(A37) yields

    $$\begin{aligned} r^{(1, j)}_{K-k}= & {} \frac{(K-k+1)\mu }{\theta }h^{(1, k-1)}_{0}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) =h^{(1, k)}_{0}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) , \\{} & {} for \ K-C+2\le k\le K-1;\\ r^{(1, j)}_{K-k}= & {} \frac{C\mu +(K-k+1-C)\gamma }{\theta }h^{(1, k-1)}_{0}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) =h^{(1, k)}_{0}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) , \\{} & {} for \ k=K-C+1;\\ r^{(1, j)}_{K-k}= & {} \frac{C\mu +(K-k+1-C)\gamma }{\theta }h^{(1, k-1)}_{0}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) =h^{(1, k)}_{0}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) ,\\{} & {} for \ k=K-C;\\ r^{(1, j)}_{K-k}= & {} \frac{C\mu +(K-k+1-C)\gamma }{\theta }h^{(1, k-1)}_{0}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) =h^{(1, k)}_{0}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) , \\{} & {} for \ 2\le k\le K-C-1; \end{aligned}$$

    where

    $$\begin{aligned} h^{(1, k)}_{0}=\frac{\lambda }{\theta }\prod ^{k-1}_{i=1}\frac{C\mu +(K-i-C)\gamma }{\theta }, \ for \ 2\le k\le K-C+1; \end{aligned}$$

    and

    $$\begin{aligned} h^{(1, k)}_{0}=\frac{\lambda }{\theta }\prod ^{K-C}_{i=1}\frac{C\mu +(K-i-C)\gamma }{\theta }\prod ^{k-1}_{i=K-C+1}\frac{(K-i)\mu }{\theta }, \ for \ K-C+2\le k\le K-1. \end{aligned}$$

    Therefore, it follows from mathematical induction that (25) and (26) are true for \(k= 1, 2,\ldots , K\).

  • The case \(k=K\):

    Theorem 5 and (25) with \(k=K-1\) and (26) yield

    $$\begin{aligned} r^{(0, j)}_{1}= & {} h^{(0, K-1)}_{0}\frac{1}{j^{K-1}}+O\left( \frac{1}{j^{K}}\right) ,\\ r^{(1, j)}_{1}= & {} h^{(1, K-1)}_{0}\frac{1}{j^{K-1}}+O\left( \frac{1}{j^{K}}\right) ,\\ r^{(0, j)}_{0}= & {} h^{(0, K)}_{0}\frac{1}{j^{K}}+o\left( \frac{1}{j^{K}}\right) \\= & {} O\left( \frac{1}{j^{K}}\right) ,\\ r^{(1, j)}_{0}= & {} h^{(1, K)}_{0}\frac{1}{j^{K}}+o\left( \frac{1}{j^{K}}\right) \\= & {} O\left( \frac{1}{j^{K}}\right) . \end{aligned}$$

    Substituting the above formulae into (A3) and (A4), we obtain

    $$\begin{aligned} r^{(0, j)}_{0}= & {} \frac{\mu }{\theta }h^{(0, K-1)}_{0}\frac{1}{j^{K}}+O\left( \frac{1}{j^{K+1}}\right) \\= & {} h^{(0, K)}_{0}\frac{1}{j^{K}}+O\left( \frac{1}{j^{K+1}}\right) ,\\ r^{(1, j)}_{0}= & {} \frac{\mu }{\theta }h^{(1, K-1)}_{0}\frac{1}{j^{K}}+O\left( \frac{1}{j^{K+1}}\right) \\= & {} h^{(1, K)}_{0}\frac{1}{j^{K}}+O\left( \frac{1}{j^{K+1}}\right) . \end{aligned}$$
  • The case \(k=0\):

    From (25) with \(k=1\) and (26), we obtain

    $$\begin{aligned} (j+1)r^{(0, j)}_{K-1}= & {} h^{(0, 1)}_{0}+O\left( \frac{1}{j}\right) ,\\ (j+1)r^{(1, j)}_{K-1}= & {} h^{(1, 1)}_{0}+O\left( \frac{1}{j}\right) . \end{aligned}$$

    Substituting the above two formulae into (A28) and (A29) yields,

    $$\begin{aligned} \left( \lambda +C\mu +(K-C)\gamma \right) r^{(0, j)}_{K}= & {} \lambda _{1} r^{(0, j)}_{K-1}+\lambda _{2} r^{(0, j)}_{K-1}+\lambda r^{(0, j)}_{K}+O\left( \frac{1}{j}\right) ,\\ \left( \lambda +C\mu +(K-C)\gamma \right) r^{(1, j)}_{K}= & {} \lambda _{1} r^{(1, j)}_{K-1}+\lambda _{2} r^{(1, j)}_{K-1}+\lambda r^{(1, j)}_{K}+\lambda +O\left( \frac{1}{j}\right) . \end{aligned}$$

    Deleting \(\lambda r^{(0, j)}_{K}\) and \(\lambda r^{(1, j)}_{K}\) from both sides of the above formulae, we obtain

    $$\begin{aligned} \left( C\mu +(K-C)\gamma \right) r^{(0, j)}_{K}= & {} \lambda _{1} r^{(0, j)}_{K-1}+\lambda _{2} r^{(0, j)}_{K-1}+O\left( \frac{1}{j}\right) ,\\ \left( C\mu +(K-C)\gamma \right) r^{(1, j)}_{K}= & {} \lambda _{1} r^{(1, j)}_{K-1}+\lambda _{2} r^{(1, j)}_{K-1}+\lambda +O\left( \frac{1}{j}\right) . \end{aligned}$$

    From the result for \(k=1\), we obtain

    $$\begin{aligned} r^{(0, j)}_{K}= & {} h^{(0, 0)}_{0}+O\left( \frac{1}{j}\right) ,\\ r^{(1, j)}_{K}= & {} h^{(0, 1)}_{0}+O\left( \frac{1}{j}\right) . \end{aligned}$$

\(\square \)

1.3 A.3 Proof of Theorem 7

Proof

We prove Theorem 7 using mathematical induction. First, we show that Theorem 7 is true for \(m=1\).

  • The case \(k=1\):

    From Theorem 4, we have

    $$\begin{aligned} r^{(0, j)}_{K-1}= & {} \frac{\lambda _{2}}{j\theta }-\sum ^{K-2}_{i=0}r^{(0, j)}_{i},\\ r^{(1, j)}_{K-1}= & {} \frac{\lambda }{j\theta }-\sum ^{K-2}_{i=0}r^{(1, j)}_{i}. \end{aligned}$$

    Theorem 6 yields

    $$\begin{aligned} r^{(0, j)}_{K-2}= & {} h^{(0, 2)}_{0}\frac{1}{j^{2}}+O\left( \frac{1}{j^{3}}\right) ,\\ r^{(1, j)}_{K-2}= & {} h^{(1, 2)}_{0}\frac{1}{j^{2}}+O\left( \frac{1}{j^{3}}\right) ,\\ \sum ^{K-3}_{i=0}r^{(0, j)}_{i}= & {} O\left( \frac{1}{j^{3}}\right) ,\\ \sum ^{K-3}_{i=0}r^{(1, j)}_{i}= & {} O\left( \frac{1}{j^{3}}\right) . \end{aligned}$$

    Thus

    $$\begin{aligned} r^{(0, j)}_{K-1}= & {} h^{(0, 1)}_{0}\frac{1}{j}-h^{(0, 1)}_{1}\frac{1}{j^{2}}+O\left( \frac{1}{j^{3}}\right) ,\\ r^{(1, j)}_{K-1}= & {} h^{(1, 1)}_{0}\frac{1}{j}-h^{(1, 1)}_{1}\frac{1}{j^{2}}+O\left( \frac{1}{j^{3}}\right) , \end{aligned}$$

    where \(h^{(0, 1)}_{1}=h^{(0, 2)}_{0}\) and \(h^{(1, 1)}_{1}=h^{(1, 2)}_{0}\).

  • The case \(k=2, 3,\ldots , K-1\):

    Assuming that (27) and (28) in Theorem 7 are true for \(r^{(0, j)}_{K-n}\) and \(r^{(1, j)}_{K-n}\) with \(n= 1, 2,\ldots , k-1\), we prove that they are also true for \(n=k\).

    Using the assumption of mathematical induction and Theorem 6, we obtain

    $$\begin{aligned} r^{(0, j)}_{K-k-1}= & {} h^{(0, k+1)}_{0}\frac{1}{j^{k+1}}+O\left( \frac{1}{j^{k+2}}\right) ,\\ r^{(0, j)}_{K-k}= & {} h^{(0, k)}_{0}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) ,\\ r^{(0, j)}_{K-k+1}= & {} h^{(0, k-1)}_{0}\frac{1}{j^{k-1}}+O\left( \frac{1}{j^{k}}\right) \\= & {} h^{(0, k-1)}_{0}\frac{1}{j^{k-1}}-h^{(0, k-1)}_{1}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) ,\\ (j+1)r^{(0, j+1)}_{K-k-1}= & {} h^{(0, k+1)}_{0}\frac{1}{(j+1)^{k}}+O\left( \frac{1}{j^{k+1}}\right) \\= & {} h^{(0, k+1)}_{0}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) ,\\ (j+1)r^{(1, j+1)}_{K-k-1}= & {} h^{(1, k+1)}_{0}\frac{1}{(j+1)^{k}}+O\left( \frac{1}{j^{k+1}}\right) \\= & {} h^{(1, k+1)}_{0}\frac{1}{j^{k}}+O\left( \frac{1}{j^{k+1}}\right) ,\\ r^{(0, j)}_{K-1}= & {} h^{(0, 1)}_{0}\frac{1}{j}-h^{(0, 1)}_{1}\frac{1}{j^{2}}+O\left( \frac{1}{j^{3}}\right) ,\\ r^{(0, j)}_{K}= & {} 0+O\left( \frac{1}{j}\right) . \end{aligned}$$

    Substituting these formulae into (A30)–(A33), we obtain

    $$\begin{aligned} r^{(0, j)}_{K-k}= & {} \frac{(K-k+1)\mu }{j^{k}\theta }h^{(0, k-1)}_{0}-\left\{ \frac{(K-k+1)\mu }{j^{k+1}\theta }h^{(0, k-1)}_{1}+\frac{(\lambda +(K-k)\mu }{j^{k+1}\theta }h^{(0, k)}_{0}\right\} \\{} & {} +O\left( \frac{1}{j^{k+2}}\right) \\= & {} h^{(0, k)}_{0}\frac{1}{j^{k}}-h^{(0, k)}_{1}\frac{1}{j^{k+1}}+O\left( \frac{1}{j^{k+2}}\right) , \\{} & {} \quad for \ K-C+2\le k\le K-1; \end{aligned}$$

    where

    $$\begin{aligned} h^{(0, k)}_{1}= & {} \frac{(K-k+1)\mu }{\theta }h^{(0, k-1)}_{1}+\frac{\lambda +(K-k)\mu }{\theta }h^{(0, k)}_{0},\\ r^{(0, j)}_{K-k}= & {} \frac{(K-k+1)\mu }{j^{k}\theta }h^{(0, k-1)}_{0}-\left\{ \frac{(K-k+1)\mu }{j^{k+1}\theta }h^{(0, k-1)}_{1}+\frac{(\lambda _{1}+(K-k)\mu }{j^{k+1}\theta }h^{(0, k)}_{0}\right\} \\{} & {} \quad +O\left( \frac{1}{j^{k+2}}\right) =h^{(0, k)}_{0}\frac{1}{j^{k}}-h^{(0, k)}_{1}\frac{1}{j^{k+1}}+O\left( \frac{1}{j^{k+2}}\right) , \\{} & {} for \ k=K-C+1; \end{aligned}$$

    where

    $$\begin{aligned} h^{(0, k)}_{1}= & {} \frac{(K-k+1)\mu }{\theta }h^{(0, k-1)}_{1}+\frac{\lambda _{1}+(K-k)\mu }{\theta }h^{(0, k)}_{0},\\ r^{(0, j)}_{K-k}= & {} \frac{C\mu +(K-k+1-C)\gamma }{j^{k}\theta }h^{(0, k-1)}_{0}-(\frac{C\mu +(K-k+1-C)\gamma }{j^{k+1}\theta }h^{(0, k-1)}_{1}\\{} & {} +\frac{(\lambda +C\mu +(K-k-C)\gamma }{j^{k+1}\theta }h^{(0, k)}_{0})+O\left( \frac{1}{j^{k+2}}\right) \\= & {} h^{(0, k)}_{0}\frac{1}{j^{k}}-h^{(0, k)}_{1}\frac{1}{j^{k+1}}+O\left( \frac{1}{j^{k+2}}\right) , \\{} & {} for \ k=K-C; \end{aligned}$$

    where

    $$\begin{aligned} h^{(0, k)}_{1}= & {} \frac{C\mu +(K-k+1-C)\gamma }{\theta }h^{(0, k-1)}_{1}+\frac{\lambda +C\mu +(K-k-C)\gamma }{\theta }h^{(0, k)}_{0},\\ r^{(0, j)}_{K-k}= & {} \frac{C\mu +(K-k+1-C)\gamma }{j^{k}\theta }h^{(0, k-1)}_{0}\\{} & {} -\left\{ \frac{C\mu +(K-k+1-C)\gamma }{j^{k+1}\theta }h^{(0, k-1)}_{1}+\frac{(\lambda +C\mu +(K-k-C)\gamma }{j^{k+1}\theta }h^{(0, k)}_{0}\right\} \\ +O\left( \frac{1}{j^{k+2}}\right)= & {} h^{(0, k)}_{0}\frac{1}{j^{k}}-h^{(0, k)}_{1}\frac{1}{j^{k+1}}+O\left( \frac{1}{j^{k+2}}\right) , \\{} & {} for \ 2\le k\le K-C-1; \end{aligned}$$

    where

    $$\begin{aligned} h^{(0, k)}_{1}=\frac{C\mu +(K-k+1-C)\gamma }{\theta }h^{(0, k-1)}_{1}+\frac{\lambda +C\mu +(K-k-C)\gamma }{\theta }h^{(0, k)}_{0}. \end{aligned}$$

    Similarly, using the same methodology, we obtain

    $$\begin{aligned} r^{(1, j)}_{K-k}= & {} \frac{(K-k+1)\mu }{j^{k}\theta }h^{(1, k-1)}_{0}-\left( \frac{(K-k+1)\mu }{j^{k+1}\theta }h^{(1, k-1)}_{1}+\frac{(\lambda +(K-k)\mu )}{j^{k+1}\theta }h^{(1, k)}_{0}\right. \\{} & {} \left. -\frac{h^{(1, 0)}_{0}h^{(1, k+1)}_{0}}{j^{k+1}}\right) +O\left( \frac{1}{j^{k+2}}\right) =h^{(1, k)}_{0}\frac{1}{j^{k}}-h^{(1, k)}_{1}\frac{1}{j^{k+1}}+O\left( \frac{1}{j^{k+2}}\right) , \\{} & {} \quad for \ K-C+2\le k\le K-1; \end{aligned}$$

    where

    $$\begin{aligned} h^{(1, k)}_{1}= & {} \frac{(K-k+1)\mu }{\theta }h^{(1, k-1)}_{1}+\frac{\lambda +(K-k)\mu }{\theta }h^{(1, k)}_{0}-h^{(1, 0)}_{0}h^{(1, k+1)}_{0},\\ r^{(1, j)}_{K-k}= & {} \frac{(K-k+1)\mu }{j^{k}\theta }h^{(1, k-1)}_{0}-\left( \frac{(K-k+1)\mu }{j^{k+1}\theta }h^{(1, k-1)}_{1}+\frac{(\lambda _{1}+(K-k)\mu )}{j^{k+1}\theta }h^{(1, k)}_{0}\right. \\{} & {} \left. -\frac{h^{(1, 0)}_{0}h^{(1, k+1)}_{0}}{j^{k+1}}\right) +O\left( \frac{1}{j^{k+2}}\right) =h^{(1, k)}_{0}\frac{1}{j^{k}}-h^{(1, k)}_{1}\frac{1}{j^{k+1}}+O\left( \frac{1}{j^{k+2}}\right) , \\{} & {} for \ k=K-C+1; \end{aligned}$$

    where

    $$\begin{aligned} h^{(1, k)}_{1}= & {} \frac{(K-k+1)\mu }{\theta }h^{(1, k-1)}_{1}+\frac{\lambda _{1}+(K-k)\mu }{\theta }h^{(1, k)}_{0}-h^{(1, 0)}_{0}h^{(1, k+1)}_{0},\\ r^{(1, j)}_{K-k}= & {} \frac{C\mu +(K-k+1-C)\gamma }{j^{k}\theta }h^{(1, k-1)}_{0}\\{} & {} -\left( \frac{C\mu +(K-k+1-C)\gamma }{j^{k+1}\theta }h^{(1, k-1)}_{1}+\frac{(\lambda +C\mu +(K-k-C)\gamma )}{j^{k+1}\theta }h^{(1, k)}_{0}\right. \\{} & {} \left. -\frac{h^{(1, 0)}_{0}h^{(1, k+1)}_{0}}{j^{k+1}}\right) +O\left( \frac{1}{j^{k+2}}\right) =h^{(1, k)}_{0}\frac{1}{j^{k}}-h^{(1, k)}_{1}\frac{1}{j^{k+1}}+O\left( \frac{1}{j^{k+2}}\right) , \\{} & {} for \ k=K-C; \end{aligned}$$

    where

    $$\begin{aligned} h^{(1, k)}_{1}= & {} \frac{C\mu +(K-k+1-C)\gamma }{\theta }h^{(1, k-1)}_{1}+\frac{\lambda +C\mu +(K-k-C)\gamma }{\theta }h^{(1, k)}_{0}\\{} & {} -h^{(1, 0)}_{0}h^{(1, k+1)}_{0}, \\{} & {} for \ 2\le k\le K-C-1. \end{aligned}$$
  • The case \(k=K\):

    Equations (27) and (28) with \(k=K-1\) and Theorem 6 yield

    $$\begin{aligned} r^{(0, j)}_{1}= & {} h^{(0, K-1)}_{0}\frac{1}{j^{K-1}}-h^{(0, K-1)}_{1}\frac{1}{j^{K}}+O\left( \frac{1}{j^{K+1}}\right) ,\\ r^{(1, j)}_{1}= & {} h^{(1, K-1)}_{0}\frac{1}{j^{K-1}}-h^{(1, K-1)}_{1}\frac{1}{j^{K}}+O\left( \frac{1}{j^{K+1}}\right) ,\\ r^{(0, j)}_{0}= & {} h^{(0, K)}_{0}\frac{1}{j^{K}}+o\left( \frac{1}{j^{K}}\right) =O\left( \frac{1}{j^{K}}\right) ,\\ r^{(1, j)}_{0}= & {} h^{(1, K)}_{0}\frac{1}{j^{K}}+o\left( \frac{1}{j^{K}}\right) =O\left( \frac{1}{j^{K}}\right) . \end{aligned}$$

    Thus, (A3) and (A4) are written as follows

    $$\begin{aligned} r^{(0, j)}_{0}= & {} \frac{\mu }{j^{K}\theta }h^{(0, K-1)}_{0}-\left\{ \frac{\lambda h^{(0, K)}_{0}+\mu h^{(0, K-1)}_{1}}{j^{K+1}\theta }\right\} +O\left( \frac{1}{j^{K+2}}\right) \\= & {} h^{(0, K)}_{0}\frac{1}{j^{K}}-h^{(0, K)}_{1}\frac{1}{j^{K+1}}+O\left( \frac{1}{j^{K+2}}\right) , \end{aligned}$$

    where

    $$\begin{aligned} h^{(0, K)}_{1}= & {} \frac{\lambda h^{(0, K)}_{0}+\mu h^{(0, K-1)}_{1}}{\theta },\\ r^{(1, j)}_{0}= & {} \frac{\mu }{j^{K}\theta }h^{(1, K-1)}_{0}-\left\{ \frac{\lambda h^{(1, K)}_{0}+\mu h^{(1, K-1)}_{1}}{j^{K+1}\theta }\right\} +O\left( \frac{1}{j^{K+2}}\right) \\= & {} h^{(1, K)}_{0}\frac{1}{j^{K}}-h^{(1, K)}_{1}\frac{1}{j^{K+1}}+O\left( \frac{1}{j^{K+2}}\right) , \end{aligned}$$

    where

    $$\begin{aligned} h^{(1, K)}_{1}=\frac{\lambda h^{(1, K)}_{0}+\mu h^{(1, K-1)}_{1}}{\theta }. \end{aligned}$$
  • The case \(k=0\):

    We use the same methodology as in Theorem 6. Equations (27) and (28) with \(k=1\) and Theorem 6 yield

    $$\begin{aligned} r^{(0, j)}_{K-1}= & {} h^{(0, 1)}_{0}\frac{1}{j}-h^{(0, 1)}_{1}\frac{1}{j^{2}}+O\left( \frac{1}{j^{3}}\right) ,\\ r^{(1, j)}_{K-1}= & {} h^{(1, 1)}_{0}\frac{1}{j}-h^{(1, 1)}_{1}\frac{1}{j^{2}}+O\left( \frac{1}{j^{3}}\right) ,\\ r^{(0, j)}_{K}= & {} h^{(0, 0)}_{0}+O\left( \frac{1}{j}\right) =0+O\left( \frac{1}{j}\right) ,\\ r^{(1, j)}_{K}= & {} h^{(1, 0)}_{0}+O\left( \frac{1}{j}\right) =\frac{\lambda }{C\mu +(K-C)\gamma }+O\left( \frac{1}{j}\right) . \end{aligned}$$

    Thus, (16) and (21) are written as follows

    $$\begin{aligned} r^{(0, j)}_{K}= & {} 0+\frac{1}{j}\left\{ \frac{\lambda _{1} h^{(0, 1)}_{0}+\theta h^{(0, 1)}_{0}h^{(0, 1)}_{0}}{C\mu +(K-C)\gamma }\right\} +O\left( \frac{1}{j^{2}}\right) \\= & {} h^{(0, 0)}_{0}-h^{(0, 0)}_{1}\frac{1}{j}+O\left( \frac{1}{j^{2}}\right) ,\\ r^{(1, j)}_{K}= & {} \frac{\lambda }{C\mu +(K-C)\gamma }+\frac{1}{j}\left\{ \frac{\lambda _{1} h^{(1, 1)}_{0}+\theta h^{(0, 1)}_{0}h^{(1, 1)}_{0}-\theta h^{(1, 0)}_{0}h^{(1, 1)}_{1}}{C\mu +(K-C)\gamma }\right\} +O\left( \frac{1}{j^{2}}\right) \\= & {} h^{(1, 0)}_{0}-h^{(1, 0)}_{1}\frac{1}{j}+O\left( \frac{1}{j^{2}}\right) . \end{aligned}$$

    Therefore, Theorem 7 is established for \(m=1\) and \(k=0, 1, C,\ldots , K\).

    Next, assuming that Theorem 7 is true for \(m-1\) (m terms expansion), we prove that it is also true for m (\(m+1\) terms expansion).

  • The case \(k=1\):

    Theorem 4 and mathematical induction yield

    $$\begin{aligned} r^{(0, j)}_{K-1}= & {} \frac{\lambda _{2}}{j\theta }-\sum ^{K}_{k=2}r^{(0, j)}_{K-k}\\= & {} \frac{\lambda _{2}}{j\theta }-\sum ^{K}_{k=2}\left\{ \sum ^{m-1}_{n=0}h^{(0, k)}_{n}(-1)^{n}\frac{1}{j^{k+n}}+O\left( \frac{1}{j^{k+m}}\right) \right\} \\= & {} \frac{\lambda _{2}}{j\theta }-\sum ^{K}_{k=2}\sum ^{m-1}_{n=0}h^{(0, k)}_{n}(-1)^{n}\frac{1}{j^{k+n}}+O\left( \frac{1}{j^{m+2}}\right) \\= & {} \sum ^{m}_{i=0}h^{(0, 1)}_{i}(-1)^{i}\frac{1}{j^{1+i}}+O\left( \frac{1}{j^{m+2}}\right) , \end{aligned}$$

    where

    $$\begin{aligned} h^{(0, 1)}_{i}=\sum ^{min(K, i+1)}_{n=2}h^{(0, n)}_{i+1-n}(-1)^{n}. \end{aligned}$$

    Similarly, we have

    $$\begin{aligned} r^{(1, j)}_{K-1}= & {} \frac{\lambda }{j\theta }-\sum ^{K}_{k=2}r^{(1, j)}_{K-k}\\= & {} \frac{\lambda }{j\theta }-\sum ^{K}_{k=2}\left\{ \sum ^{m-1}_{n=0}h^{(1, k)}_{n}(-1)^{n}\frac{1}{j^{k+n}}+O\left( \frac{1}{j^{k+m}}\right) \right\} \\= & {} \frac{\lambda }{j\theta }-\sum ^{K}_{k=2}\sum ^{m-1}_{n=0}h^{(1, k)}_{n}(-1)^{n}\frac{1}{j^{k+n}}+O\left( \frac{1}{j^{m+2}}\right) \\= & {} \sum ^{m}_{i=0}h^{(1, 1)}_{i}(-1)^{i}\frac{1}{j^{1+i}}+O\left( \frac{1}{j^{m+2}}\right) , \end{aligned}$$

    where

    $$\begin{aligned} h^{(1, 1)}_{i}=\sum ^{min(K, i+1)}_{n=2}h^{(1, n)}_{i+1-n}(-1)^{n}. \end{aligned}$$
  • The case \(k=2, 3,\ldots , K-1\):

    Assuming that (27) and (28) in Theorem 7 are true for \(r^{(0, j)}_{K-n}\) and \(r^{(1, j)}_{K-n}\) with \(n= 1,\ldots , k-1\), we prove that they are also true for \(n=k\). Applying the assumption of mathematical induction and (27) for \(k=1\) yields

    $$\begin{aligned} r^{(0, j)}_{K-k-1}= & {} \sum ^{m-1}_{i=0}h^{(0, k+1)}_{i}(-1)^{i}\frac{1}{j^{k+i+1}}+O\left( \frac{1}{j^{k+m+1}}\right) ,\\ r^{(0, j)}_{K-k}= & {} \sum ^{m-1}_{i=0}h^{(0, k)}_{i}(-1)^{i}\frac{1}{j^{k+i}}+O\left( \frac{1}{j^{k+m}}\right) ,\\ r^{(0, j)}_{K-k+1}= & {} \sum ^{m}_{i=0}h^{(0, k-1)}_{i}(-1)^{i}\frac{1}{j^{k+i-1}}+O\left( \frac{1}{j^{k+m}}\right) ,\\ r^{(0, j)}_{K-1}= & {} \sum ^{m}_{i=0}h^{(0, 1)}_{i}(-1)^{i}\frac{1}{j^{1+i}}+O\left( \frac{1}{j^{m+2}}\right) ,\\ r^{(0, j)}_{K}= & {} \sum ^{m-1}_{i=0}h^{(0, 0)}_{i}(-1)^{i}\frac{1}{j^{i}}+O\left( \frac{1}{j^{m}}\right) ,\\ (j+1)r^{(0, j+1)}_{K-k-1}= & {} \sum ^{m-1}_{i=0}h^{(0, k+1)}_{i}(-1)^{i}\frac{1}{(j+1)^{k+i}}+O\left( \frac{1}{j^{k+m}}\right) \\= & {} \sum ^{m-1}_{n=0}\phi ^{(0, k)}_{n}\frac{1}{j^{k+n}}+O\left( \frac{1}{j^{k+m}}\right) ,\\ (j+1)r^{(1, j+1)}_{K-k-1}= & {} \sum ^{m-1}_{i=0}\phi ^{(1, k+1)}_{i}(-1)^{i}\frac{1}{(j+1)^{k+i}}+O\left( \frac{1}{j^{k+m}}\right) \\= & {} \sum ^{m-1}_{n=0}\phi ^{(1, k)}_{n}\frac{1}{j^{k+n}}+O\left( \frac{1}{j^{k+m}}\right) . \end{aligned}$$

    Substituting these formulae into (A30)–(A33) and attracting the coefficient of \(\frac{1}{j^{k+m}}\) of (A30)-(A33) and arranging the result, we obtain

    $$\begin{aligned} h^{(0, k)}_{m}= & {} \frac{\lambda }{\theta }h^{(0, k+1)}_{m-2}+\frac{\lambda +(K-k)\mu }{\theta }h^{(0, k)}_{m-1}+\frac{(K-k+1)\mu }{\theta }h^{(0, k-1)}_{m}\\{} & {} +\sum ^{m-2}_{n=0}\phi ^{(0, k)}_{n}(-1)^{n}h^{(0, 1)}_{m-n-2}+\sum ^{m-1}_{n=0}\phi ^{(1, k)}_{n}(-1)^{n+1}h^{(0, 0)}_{m-n-1}, \\{} & {} \text {for} \ K-C+2\le k\le K-1;\\ h^{(0, k)}_{m}= & {} \frac{\lambda }{\theta }h^{(0, k+1)}_{m-2}+\frac{\lambda _{1}+(K-k)\mu }{\theta }h^{(0, k)}_{m-1}+\frac{(K-k+1)\mu }{\theta }h^{(0, k-1)}_{m}\\{} & {} +\sum ^{m-2}_{n=0}\phi ^{(0, k)}_{n}(-1)^{n}h^{(0, 1)}_{m-n-2}+\sum ^{m-1}_{n=0}\phi ^{(1, k)}_{n}(-1)^{n+1}h^{(0, 0)}_{m-n-1}, \\{} & {} \text {for} \ k= K-C+1;\\ h^{(0, k)}_{m}= & {} \frac{\lambda _{1}}{\theta }h^{(0, k+1)}_{m-2}+\frac{\lambda +C\mu +(K-k-C)\gamma }{\theta }h^{(0, k)}_{m-1}+\frac{C\mu +(K-k+1-C)\gamma }{\theta }h^{(0, k-1)}_{m}\\{} & {} +\sum ^{m-2}_{n=0}\phi ^{(0, k)}_{n}(-1)^{n}h^{(0, 1)}_{m-n-2}+\sum ^{m-1}_{n=0}\phi ^{(1, k)}_{n}(-1)^{n+1}h^{(0, 0)}_{m-n-1},\\{} & {} \text {for} \ k= K-C;\\ h^{(0, k)}_{m}= & {} \frac{\lambda }{\theta }h^{(0, k+1)}_{m-2}+\frac{\lambda +C\mu +(K-k-C)\gamma }{\theta }h^{(0, k)}_{m-1}+\frac{C\mu +(K-k+1-C)\gamma }{\theta }h^{(0, k-1)}_{m}\\{} & {} +\sum ^{m-2}_{n=0}\phi ^{(0, k)}_{n}(-1)^{n}h^{(0, 1)}_{m-n-2}+\sum ^{m-1}_{n=0}\phi ^{(1, k)}_{n}(-1)^{n+1}h^{(0, 0)}_{m-n-1}, \\{} & {} \text {for} \ 2\le k\le K-C-1. \end{aligned}$$

    Similarly, using the same methodology, we obtain

    $$\begin{aligned} h^{(1, k)}_{m}= & {} \frac{\lambda }{\theta }h^{(1, k+1)}_{m-2}+\frac{\lambda +(K-k)\mu }{\theta }h^{(1, k)}_{m-1}+\frac{(K-k+1)\mu }{\theta }h^{(1, k-1)}_{m}\\{} & {} +\sum ^{m-2}_{n=0}\phi ^{(0, k)}_{n}(-1)^{n}h^{(1, 1)}_{m-n-2}+\sum ^{m-1}_{n=0}\phi ^{(1, k)}_{n}(-1)^{n+1}h^{(1, 0)}_{m-n-1}, \\{} & {} \text {for} \ K-C+2\le k\le K-1;\\ h^{(1, k)}_{m}= & {} \frac{\lambda }{\theta }h^{(1, k+1)}_{m-2}+\frac{\lambda _{1}+(K-k)\mu }{\theta }h^{(1, k)}_{m-1}+\frac{(K-k+1)\mu }{\theta }h^{(1, k-1)}_{m}\\{} & {} +\sum ^{m-2}_{n=0}\phi ^{(0, k)}_{n}(-1)^{n}h^{(1, 1)}_{m-n-2}+\sum ^{m-1}_{n=0}\phi ^{(1, k)}_{n}(-1)^{n+1}h^{(1, 0)}_{m-n-1}, \\{} & {} \text {for} \ k= K-C+1;\\ h^{(1, k)}_{m}= & {} \frac{\lambda _{1}}{\theta }h^{(1, k+1)}_{m-2}+\frac{\lambda +C\mu +(K-k-C)\gamma }{\theta }h^{(1, k)}_{m-1}+\frac{C\mu +(K-k+1-C)\gamma }{\theta }h^{(1, k-1)}_{m}\\{} & {} +\sum ^{m-2}_{n=0}\phi ^{(0, k)}_{n}(-1)^{n}h^{(1, 1)}_{m-n-2}+\sum ^{m-1}_{n=0}\phi ^{(1, k)}_{n}(-1)^{n+1}h^{(1, 0)}_{m-n-1}, \\{} & {} \text {for} \ k= K-C;\\ h^{(1, k)}_{m}= & {} \frac{\lambda }{\theta }h^{(1, k+1)}_{m-2}+\frac{\lambda +C\mu +(K-k-C)\gamma }{\theta }h^{(1, k)}_{m-1}+\frac{C\mu +(K-k+1-C)\gamma }{\theta }h^{(1, k-1)}_{m}\\{} & {} +\sum ^{m-2}_{n=0}\phi ^{(0, k)}_{n}(-1)^{n}h^{(1, 1)}_{m-n-2}+\sum ^{m-1}_{n=0}\phi ^{(1, k)}_{n}(-1)^{n+1}h^{(1, 0)}_{m-n-1}, \\{} & {} \text {for} \ 2\le k\le K-C-1. \end{aligned}$$

    Thus, we obtain the result for the case \(k=2, 3,\ldots , K-1\).

  • The case \(k=K\): Using Theorem 6, (27) and (28) with \(k=K-1\), we obtain

    $$\begin{aligned} r^{(0, j)}_{1}= & {} \sum ^{m}_{i=0}h^{(0, K-1)}_{i}(-1)^{i}\frac{1}{j^{K-1+i}}+O\left( \frac{1}{j^{K+m}}\right) ,\\ r^{(1, j)}_{1}= & {} \sum ^{m}_{i=0}h^{(1, K-1)}_{i}(-1)^{i}\frac{1}{j^{K-1+i}}+O\left( \frac{1}{j^{K+m}}\right) ,\\ r^{(0, j)}_{0}= & {} \sum ^{m-1}_{i=0}h^{(0, K)}_{i}(-1)^{i}\frac{1}{j^{K+i}}+O\left( \frac{1}{j^{K+m}}\right) ,\\ r^{(1, j)}_{0}= & {} \sum ^{m-1}_{i=0}h^{(1, K)}_{i}(-1)^{i}\frac{1}{j^{K+i}}+O\left( \frac{1}{j^{K+m}}\right) . \end{aligned}$$

    Attracting the coefficient of \(\frac{1}{j^{K+m}}\) in (A3) and (A4) and arranging the result yields

    $$\begin{aligned} h^{(0, K)}_{m}= & {} \frac{\lambda }{\theta }h^{(0, K)}_{m-1}+\frac{\mu }{\theta }h^{(0, K-1)}_{m},\\ h^{(1, K)}_{m}= & {} \frac{\lambda }{\theta }h^{(1, K)}_{m-1}+\frac{\mu }{\theta }h^{(1, K-1)}_{m}. \end{aligned}$$

    Thus, we obtain the desired result for the case \(k=K\).

  • The case \(k=0\):

    We can prove Theorem 6 using the same methodology. Equations (27) and (28) with \(k=1\) and Theorem 6 yield

    $$\begin{aligned} r^{(0, j)}_{K-1}= & {} \sum ^{m}_{i=0}h^{(0, 1)}_{i}(-1)^{i}\frac{1}{j^{1+i}}+O\left( \frac{1}{j^{m+2}}\right) ,\\ r^{(1, j)}_{K-1}= & {} \sum ^{m}_{i=0}h^{(1, 1)}_{i}(-1)^{i}\frac{1}{j^{1+i}}+O\left( \frac{1}{j^{m+2}}\right) ,\\ r^{(0, j)}_{K}= & {} \sum ^{m-1}_{i=0}h^{(0, 0)}_{i}(-1)^{i}\frac{1}{j^{i}}+O\left( \frac{1}{j^{m}}\right) ,\\ r^{(1, j)}_{K}= & {} \sum ^{m-1}_{i=0}h^{(1, 0)}_{i}(-1)^{i}\frac{1}{j^{i}}+O\left( \frac{1}{j^{m}}\right) ,\\ (j+1)r^{(0, j+1)}_{K-1}= & {} \sum ^{m}_{i=0}h^{(0, 1)}_{i}(-1)^{i}\frac{1}{(j+1)^{i}}+O\left( \frac{1}{j^{1+m}}\right) =\sum ^{m}_{n=0}\phi ^{(0, 0)}_{n}\frac{1}{j^{n}}+O\left( \frac{1}{j^{1+m}}\right) ,\\ (j+1)r^{(1, j+1)}_{K-1}= & {} \sum ^{m}_{i=0}\phi ^{(1, 1)}_{i}(-1)^{i}\frac{1}{(j+1)^{i}}+O\left( \frac{1}{j^{1+m}}\right) =\sum ^{m}_{n=0}\phi ^{(1, 0)}_{n}\frac{1}{j^{n}}+O\left( \frac{1}{j^{1+m}}\right) . \end{aligned}$$

    Using these formulae and attracting the coefficient of \(\frac{1}{j^{m}}\) in (A28) and (A29), we obtain

    $$\begin{aligned} h^{(0, 0)}_{m}= & {} -\frac{\lambda _{1}}{C\mu +(K-C)\gamma }h^{(0, 1)}_{m-1}+\frac{\theta }{C\mu +(K-C)\gamma }\sum ^{m-1}_{n=0}\phi ^{(0, 0)}_{n}(-1)^{n+1}h^{(0, 1)}_{m-n-1}\\{} & {} +\frac{\theta }{C\mu +(K-C)\gamma }\sum ^{m}_{n=1}\phi ^{(1, 0)}_{n}(-1)^{n}h^{(0, 0)}_{m-n},\\ h^{(1, 0)}_{m}= & {} -\frac{\lambda _{1}}{C\mu +(K-C)\gamma }h^{(1, 1)}_{m-1}+\frac{\theta }{C\mu +(K-C)\gamma }\sum ^{m-1}_{n=0}\phi ^{(0, 0)}_{n}(-1)^{n+1}h^{(1, 1)}_{m-n-1}\\{} & {} +\frac{\theta }{C\mu +(K-C)\gamma }\sum ^{m}_{n=1}\phi ^{(1, 0)}_{n}(-1)^{n}h^{(1, 0)}_{m-n}. \end{aligned}$$

\(\square \)

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Zidani, N., Djellab, N. Asymptotic upper bounds for an M/M/C/K retrial queue with a guard channel and guard buffer. Math Meth Oper Res 99, 365–407 (2024). https://doi.org/10.1007/s00186-024-00865-0

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