2. The PDMP framework

Piecewise deterministic Markov processes were introduced by Davis [Reference Davis7] as a general family of non-diffusion stochastic models. These Markov processes consist of a mixture of deterministic motion and random jumps. The class of PDMPs provides a framework for studying optimization problems especially in queueing systems [Reference Breuer1,Reference Davis8] . As shown in Davis [Reference Davis7] and then extensively indicated in Davis [Reference Davis8], a PDMP can be explicitly determined by means of three parameters $(\mathfrak {X},\lambda,Q)$. Let $X(t)$ be a PDMP in a state space $\mathbb {E}$ defined as follows. Let $K$ be a countable set and $d:K\longrightarrow \mathbb {N}$ be a given function. For each $\upsilon \in K$, $M_{\upsilon }$ is an open set of $\mathbb {R}^{d(\upsilon )}$. Then, $\mathbb {E}=\bigcup _{\upsilon \in K}M_{\upsilon }=\{(\upsilon,\xi ): \upsilon \in K, \xi \in M_{\upsilon }\}$. Denote by $\mathcal {E}$ the $\sigma$-algebra generated by the Borel subsets of $\mathbb {E}$. The state of the process will be denoted by $X(t)=(\upsilon (t),\xi (t))$ and the characteristics $\mathfrak {X}$, $\lambda$ and $Q$ are defined as follows:

• • $\mathfrak {X}=\{\mathfrak {X}_{\upsilon }, \upsilon \in K\}$ is a locally Lipschitz continuous vector fields in $\mathbb {E}$ with flow functions $\phi _{\upsilon }(t,\xi )$ for each $\xi \in M_{\upsilon }$.

• • $\lambda : \mathbb {E}\rightarrow \mathbb {R}_{+}$ is the jump rate. It is assumed that this function is measurable and for all $x=(\upsilon,\xi ) \in \mathbb {E}$, there exists $\varepsilon > 0$ such that $\int _{0}^{\varepsilon } \lambda (\phi _{\upsilon }(t,\xi )) \,dt$ exists.

• • $Q:(\mathbb {E} \cup \partial ^{*} \mathbb {E} )\times \mathcal {E} \rightarrow [0,1]$, with $\partial ^{*} \mathbb {E} =\bigcup _{\upsilon \in K}\partial ^{*}M_{\upsilon }$ where $\partial ^{*}M_{\upsilon }=\{\xi ^{\prime }\in \partial M_{\upsilon }: \phi _{\upsilon }(t,\xi )=\xi ^{\prime },\text { for all } (t,\xi )\in \mathbb {R}_{+}\times M_{\upsilon }\}$, a transition measure specifying the post-jump locations with $Q(x; \{x\})=0$. Note that $\partial M_{\upsilon }$ is the boundary of $M_{\upsilon }$ and $\partial ^{*} M_{\upsilon }$ represents those boundary points at which the flow exits from $M_{\upsilon }$.

With a convenient choice of the state space $\mathbb {E}$ and parameters $\mathfrak {X}$, $\lambda$ and $Q$ it is possible to model almost all non-diffusion processes found in the literature. Several important applications were presented in Davis [Reference Davis7,Reference Davis8] . The motion of the PDMP depends on the characteristics $\mathfrak {X}$, $\lambda$ and $Q$ in the following way. Starting from a point $x=(\upsilon,\xi )\in \mathbb {E}$, we select a jump time $T_{1}$ with distribution function $P_{x}(T_{1}>t)=\mathbf {I}_{t< t_{*}(x)}\exp \{-\int _{0}^{t}\lambda (\phi _{\upsilon }(s,\xi )) \,ds\}$, where $t_{*}(x)= \inf \{t \in \mathbb {R}_{+}: \phi _{\upsilon }(t,\xi )\in \partial ^{*} M_{\upsilon } \}$. The deterministic variable $t_{*}(x)$ denotes the time until the set $\partial ^{*} \mathbb {E}$ is reached from a state $x \in \mathbb {E}$. Now select a random variable $Z_{1}$ having distribution $Q(\phi _{\upsilon }(T_{1},\xi );\cdot )$. Hence, the trajectory of $X(t)$ for $t\leq T_{1}$ is given by

$$X(t)=\begin{cases} (\upsilon,\phi_{\upsilon}(t,\xi)) & \text{for}\ t< T_{1};\\ Z_{1} & \text{for}\ t= T_{1}. \end{cases}$$

Starting from $X(T_{1})$ we now select the next inter-jump time $T_{2}-T_{1}$ and the post-jump location $X(T_{2})$ in a similar way. This gives a piecewise deterministic trajectory $(X(t))$ with jump times $T_{1},T_{2},\ldots$ and post-jump locations $Z_{1},Z_{2},\ldots$. The sequence of random variables $\{Z_{n}\}$ is the Markov chain associated with the original process $X(t)$, so that previous results on the well-known discrete-time Markov chains were used to investigate the stability and ergodicity of PDMPs, see [Reference Costa4,Reference Costa and Dufour5,Reference Dufour and Costa9] . Furthermore, it is assumed that there are only finite many jumps of $X(t)$ in any finite time interval (see assumption 3.1 in Davis [Reference Davis7] or assumption 24.4 in Davis [Reference Davis8]). Thus, if $Z_{0}$ is the initial point then the associated Markov chain $\{Z_{n}, n\geq 0\}$ has the transition measure $\mathcal {P}: \mathbb {E} \cup \partial ^{*} \mathbb {E} \times \mathcal {E} \rightarrow [0,1]$ given by:

\begin{align*} \mathcal{P}(x; A)& =\int_{0}^{t_{*}(x)}Q(\phi_{\upsilon}(s,\xi);A)\lambda(\phi_{\upsilon}(s,\xi))\exp(-\Lambda(s,x))\,ds\\ & \quad +Q(\phi_{\upsilon}(t_{*}(x),\xi);A)\exp(-\Lambda(t_{*}(x),x)). \end{align*}

where $\Lambda (t,x)=\int _{0}^{t}\lambda (\phi _{\upsilon }(s,\xi ))\,ds$ for all $x\in \mathbb {E}$ and $0\leq t\leq t_{*}(x)$.

Note that $\Lambda (t_{*}(x),x)=\infty$ whenever $t_{*}(x)=\infty$.

3. M/G/1 queue with general retrial times as a PDMP

The model considered is an M/G/1 retrial queue. Customers arrive to the system according to a Poisson process with rate $\lambda$. Each incoming customer that finds the server busy joins the orbit. Service times are i.i.d. with the same general distribution function $F$. After a random time in the orbit, each blocked customer repeats his attempt to enter service. Retrial times are i.i.d. with the same general distribution function $G$. Inter-arrivals, service periods and retrial times are assumed to be mutually independent. The model studied can be represented as a PDMP in the following way. Let $(X(t)=(\upsilon (t),\xi (t));t\in \mathbb {R}_{+})$, where $\upsilon (t)=(C(t),N(t))$ and $\xi (t)=(Y(t),\mathbf {R}(t))$, be the PDMP describing the system state at time $t$. The component $C(t)$ represents the state of the server ($C(t)$ equals 1 or 0 according as the server is busy or free), $N(t)$ is the number of the blocked customers, $Y(t)$ is the residual service time and $\mathbf {R}(t)=(R_{1}(t),R_{2}(t),\ldots,R_{N(t)}(t))$ where $R_{k}(t)$ denotes the remaining retrial time of the $k$th blocked customer for $1\leq k \leq N(t)$. The considered process is defined on the state space $\mathbb {E}=(\mathbb {E}_{0}\cup \partial ^{*}\mathbb {E}_{0}) \cup (\mathbb {E}_{1}\cup \partial ^{*}\mathbb {E}_{1})$, where $\mathbb {E}_{0}=\{(0,n,0,r_{1},r_{2},\ldots,r_{n}),\ n\in \mathbb {N},\ 0< r_{1}< r_{2}<\cdots < r_{n}\}$, $\partial ^{*}\mathbb {E}_{0}=\{(0,n,0,0,r_{2},\ldots,r_{n}),\ n\in \mathbb {N}^{*},\ 0< r_{2}<\cdots < r_{n}\}$, $\mathbb {E}_{1}=\{(1,n,y,r_{1},r_{2},\ldots,r_{n}),\ n\in \mathbb {N},\ y>0,\ 0< r_{1}< r_{2}<\cdots < r_{n}\}$ and $\partial ^{*} \mathbb {E}_{1}=\{(1,n,0,r_{1},r_{2},\ldots,r_{n}),\ n\in \mathbb {N},\ 0< r_{1}< r_{2}<\cdots < r_{n}\}$. This particular formulation of the state space $\mathbb {E}$ allows us to specify the transition measure $Q(x;\cdot )$ of $X(t)$ effortlessly and one can see that it remains suitable to our case where all blocked customers retry to capture the service simultaneously.

Define the flow function $\phi$:

$$\phi(t,x)=\begin{cases} (0,n,0,r_{1}-t,r_{2}-t,\ldots,r_{n}-t) & \text{for}\ x\in \mathbb{E}_{0};\\ (1,n,y-t,r_{1}-t,r_{2}-t,\ldots,r_{n}-t) & \text{for}\ x\in \mathbb{E}_{1}. \end{cases}$$

We also give some notations for further use. Let $\mathcal {B}_{(a,b)}$ be the $\sigma$-algebra of Borel sets on the interval $(a,b)$ and $\beta _{n}$ be the Borel $\sigma$-algebra on the set $F^{(n)}=\{(u_{1},\ldots,u_{n})\in (0,\infty )^{n} : u_{1}<\cdots < u_{n} \}$. We define the following function as well for $A\in \beta _{n}$

$$\mathbf{1}_{A}(u_{1},\ldots,u_{n})=\begin{cases} 1 & \text{if}\ (u_{1},\ldots,u_{n})\in A ,\\ 0 & \text{otherwise}. \end{cases}$$

In the same spirit of Gómez-Corral and López-García [Reference Gómez-Corral and López-García15] and Breuer[Reference Breuer1], in order to describe the jumps that can occur more clearly, we are going to introduce three transition measures $Q_{1},\ Q_{2}$ and $Q_{3}$ which reflect the transitions associated with the arrival process, the service achievement and the removal of a blocked customer from the orbit, respectively.

For states $x\in \mathbb {E}_{0}$, the transition measure $Q_{1}(x;\cdot )$ is given by

$$Q_{1}(x;\{1\}\times \{n\}\times A \times B)= F(A)\mathbf{1}_{B}(r_{1},\ldots,r_{n}),$$

where $A\in \mathcal {B}_{(0,\infty )}$ and $B\in \beta _{n}$.

The transition measure $Q_{1}(x;\cdot )$ captures the transition from state $(0,n,0,r_{1},\ldots,r_{n})$ to state $(1,n,y,r_{1},\ldots,r_{n})$ related to a new external incoming customer to the queueing system that joins the idle server immediately.

For states $x\in \partial ^{*}\mathbb {E}_{0}$, $A\in \mathcal {B}_{(0,\infty )}$ and $B\in \beta _{n-1}$, the transition measure $Q_{3}(x;\cdot )$ is given by

$$Q_{3}(x;\{1\}\times \{n-1\}\times A \times B)= F(A)\mathbf{1}_{B}(r_{2},\ldots,r_{n}).$$

This refers to the transition from state $(0,n,0,0,r_{2},\ldots,r_{n})$ to state $(1,n-1,y,r_{1}^{\prime },\ldots,r_{n-1}^{\prime })$ occurring when a blocked customer joins the idle server.

In a similar manner, we determine the transition measures $Q_{1}(x;\cdot )$ and $Q_{2}(x;\cdot )$ for states $x\in \mathbb {E}_{1}$ and $x\in \partial ^{*} \mathbb {E}_{1}$, respectively. The transition measure $Q_{1}(x;\cdot )$ is specified as follows:

1. (i) For $A\in \mathcal {B}_{(0,\infty )}$, $B\in \mathcal {B}_{ (0,r_{1})}$ and sets $C\in \beta _{n}$,

   $$Q_{1}(x;\{1\}\times \{n+1\}\times A \times B \times C)=\mathbf{1}_{A}(y)G(B)\mathbf{1}_{C}(r_{1},\ldots,r_{n}).$$

2. (ii) For $A\in \mathcal {B}_{(0,\infty )}$, sets $B\in \beta _{k}$, $C \in \mathcal {B}_{(r_{k},r_{k+1})}$ and sets $D\in \beta _{n-k}$,

   $$Q_{1}(x;\{1\}\times \{n+1\}\times A \times B \times C \times D)=\mathbf{1}_{A}(y)\mathbf{1}_{B}(r_{1},\ldots,r_{k})G(C)\mathbf{1}_{D}(r_{k+1},\ldots,r_{n}).$$

3. (iii) $A\in \mathcal {B}_{(0,\infty )}$, sets $B\in \beta _{n}$ and $C \in \mathcal {B}_{(r_{n},\infty )}$,

   $$Q_{1}(x;\{1\}\times \{n+1\}\times A \times B \times C)=\mathbf{1}_{A}(y)\mathbf{1}_{B}(r_{1},\ldots,r_{n})G(C).$$

The transition measure $Q_{1}(x;\cdot )$ captures the transition from state $(1,n,y,r_{1},\ldots,r_{n})$ to state $(1,n+1,y,r_{1}^{\prime },\ldots, r_{n}^{\prime },r_{n+1}^{\prime })$ resulting when an incoming customer finds the server busy, therefore it joins the orbit. Thus, it becomes a blocked customer and a new remaining retrial time generated from $G$ must be added to the vector $(r_{1},\ldots,r_{n})$ in its convenient place yielding to a new vector of remaining retrial times $(r_{1}^{\prime },\ldots, r_{n}^{\prime },r_{n+1}^{\prime })$ with $r_{1}^{\prime }<\cdots < r_{n}^{\prime }< r_{n+1}^{\prime }$.

Finally, the transition measure $Q_{2}(x;\cdot )$ associated with the transition from state $(1,n,0,r_{1},\ldots,r_{n})$ to state $(0,n,0,r_{1},\ldots,r_{n})$, when the server becomes idle, is given by

$$Q_{2}(x;\{0\}\times \{n\}\times\{0\} \times A)= \mathbf{1}_{A}(r_{1},\ldots,r_{n}),$$

where $A\in \beta _{n}$.

Note that, for our process, the jump rate is exactly the arrival rate $\lambda$. Hence, we have $\Lambda (t,x)=\int _{0}^{t}\lambda (\phi (s,x))\,ds=\lambda t$.

4. The associated martingales

In this section, we will derive the martingales associated with the PDMP that models our retrial queue using its extended generator.

Theorem 1. For $0\leq z_{1} \leq 1$, $0\leq z_{2}\leq 1$, $\gamma \geq 0$ and $\delta \geq 0$ , the function

(4.1)\begin{equation} z_{1}^{C(t)}z_{2}^{N(t)}\,e^{-\gamma Y(t)} \,e^{-\delta \sum_{k=1}^{N(t)}R_{k}(t)}\,e^{\theta_{C(t)}(t)} \end{equation}

with

(4.2)\begin{equation} \theta_{C(t)}(t)=\begin{cases} (\lambda-N(t)\delta)t-\ln\{\lambda z_{2}\varphi_{R}(\delta)+\gamma\}\\ \quad + \ln\{\lambda z_{1}\varphi_{S}(\delta)[e^{-(\lambda z_{2}\varphi_{R}(\delta)+\gamma)t}-1]+\lambda z_{2}\varphi_{R}(\delta)+\gamma\} & \text{for}\ C(t)=0;\\ (\lambda-\lambda z_{2}\varphi_{R}(\delta)-N(t)\delta-\gamma)t & \text{for}\ C(t)=1. \end{cases} \end{equation}

is a martingale for states $x \in \mathbb {E}_{C(t)}$, where

$$\varphi_{S}(\gamma)=\int_{0}^{\infty}e^{-\gamma y}\,dF(y)\quad \mathrm{and}\quad \varphi_{R}(\delta)=\int_{0}^{\infty}e^{-\delta r}\,dG(r).$$

Proof. The infinitesimal generator of the process $X(t)$, acting on a function $f(i,n,y,r_{1},\ldots,r_{n},t)\in \mathfrak {D}(\mathcal {G})$, is given by

(4.3)\begin{align} & \mathcal{G}f(0,n,0,r_{1},r_{2},\ldots,r_{n},t)\nonumber\\ & \quad =\lambda \int_{0}^{\infty}[f(1,n,y,r_{1},r_{2},\ldots,r_{n},t)-f(0,n,0,r_{1},r_{2},\ldots,r_{n},t)]\,dF(y)\nonumber\\ & \qquad - \sum_{k=1}^{n}\dfrac{\partial}{\partial r_{k}} f(0,n,0,r_{1},r_{2},\ldots,r_{n},t)+\dfrac{\partial}{\partial t}f(0,n,0,r_{1},r_{2},\ldots,r_{n},t).\end{align}

(4.4)\begin{align} & \mathcal{G}f(1,n,y,r_{1},r_{2},\ldots,r_{n},t)\nonumber\\ & \quad =\lambda \int_{0}^{\infty}[f(1,n+1,y,r_{1},r_{2},\ldots,r_{n},r_{n+1},t) -f(1,n,y,r_{1},r_{2},\ldots,r_{n},t)]\,dG(r_{n+1})\nonumber\\ & \qquad - \dfrac{\partial}{\partial y}f(1,n,y,r_{1},r_{2},\ldots,r_{n},t) - \sum_{k=1}^{n}\dfrac{\partial}{\partial r_{k}} f(1,n,y,r_{1},r_{2},\ldots,r_{n},t)\nonumber\\ & \qquad +\dfrac{\partial}{\partial t}f(1,n,y,r_{1},r_{2},\ldots,r_{n},t). \end{align}

where $\mathfrak {D}(\mathcal {G})$ is the domain of the generator $\mathcal {G}$ which consists of those functions $f(i,n,y,r_{1},\ldots,r_{n},t)$ that are differentiable with respect to $y,r_{1},\ldots,r_{n}$ and $t$ for all $i,n,y,r_{1},\ldots,r_{n},t$, and satisfy the boundary conditions derived from Eq. (5.4) in Davis [Reference Davis7]

(4.5)\begin{align} f(0,n,0,0,r_{2},\ldots,r_{n},t) & =\int_{0}^{\infty}f(1,n-1,y,r^{\prime}_{1},\ldots,r_{n-1}^{\prime},t)\,dF(y); \end{align}

(4.6)\begin{align} f(1,n,0,r_{1},\ldots, r_{n},t)& =f(0,n,0,r_{1},\ldots,r_{n},t); \end{align}

where $(r^{\prime }_{1},\ldots,r_{n-1}^{\prime })=(r_{2},\ldots,r_{n})$ and verify the integrability conditions

(4.7)\begin{align} & E\left\{\left|\int_{0}^{\infty}f(1,n,y,r_{1},\ldots,r_{n},t)\,dF(y) -f(0,n,0,r_{1},\ldots,r_{n},t)\right|\right\}<\infty; \end{align}

(4.8)\begin{align} & E\left\{\left|\int_{0}^{\infty}f(1,n+1,y,r_{1},\ldots,r_{n},r_{n+1},t)\,dG(r_{n+1}) -f(1,n,y,r_{1},\ldots,r_{n},t)\right|\right\}<\infty. \end{align}

Define now the function

(4.9)\begin{equation} f(i,n,y,r_{1},r_{2},\ldots,r_{n},t)=z_{1}^{i}z_{2}^{n}\,e^{-\gamma y}\,e^{-\delta \sum_{k=1}^{n}r_{k}}\,e^{\theta_{i}(t)}, \end{equation}

where $\theta _{i}(t)$ is given by Eq. (4.2) for $C(t)=i,\ i \in \{0,1\}$.

By substituting Eq. (4.9) in Eq. (4.3) for $i=0$, we get

\begin{align*} \mathcal{G}f(0,n,0,r_{1},r_{2},\ldots,r_{n},t)& =\lambda\int_{0}^{\infty}[z_{1}\,e^{-\gamma y}\,e^{\theta_{1}(t)-\theta_{0}(t)}-1]\,dF(y)+\sum_{k=1}^{n}\delta+\theta_{0}^{\prime}(t)\\ & =\lambda z_{1}\,e^{\theta_{1}(t)-\theta_{0}(t)}\varphi_{S}(\gamma)-\lambda+n\delta+\theta_{0}^{\prime}(t)\\ & =\lambda z_{1}\dfrac{(\lambda z_{2}\varphi_{R}(\delta)+\gamma)\,e^{-(\lambda z_{2}\varphi_{R}(\delta)+\gamma)}}{\lambda z_{1}\varphi_{S}(\gamma)[e^{-(\lambda z_{2}\varphi_{R}(\delta)+\gamma)}-1]+\lambda z_{2}\varphi_{R}(\delta)+\gamma}\varphi_{S}(\gamma)\\ & \quad -\lambda+n\delta+\lambda-n\delta\\ & \quad -\lambda z_{1}\varphi_{S}(\gamma)\dfrac{(\lambda z_{2}\varphi_{R}(\delta)+\gamma)\,e^{-(\lambda z_{2}\varphi_{R}(\delta)+\gamma)}}{\lambda z_{1}\varphi_{S}(\gamma)[e^{-(\lambda z_{2}\varphi_{R}(\delta)+\gamma)}-1]+\lambda z_{2}\varphi_{R}(\delta)+\gamma}\\ & =0. \end{align*}

Similarly, we substitute Eq. (4.9) in Eq. (4.4) for $i=1$.

\begin{align*} \mathcal{G}f(1,n,y,r_{1},r_{2},\ldots,r_{n},t)& =\lambda \int_{0}^{\infty}[z_{2}\,e^{-\delta r_{n+1}-1}]\,dG(r_{n+1})+\gamma+n\delta+\theta_{1}^{\prime}(t)\\ & =\lambda z_{2} \varphi_{R}(\delta)-\lambda+\gamma+n\delta+\theta_{1}^{\prime}(t)\\ & =\lambda z_{2} \varphi_{R}(\delta)-\lambda+\gamma+n\delta+\lambda -\lambda z_{2} \varphi_{R}(\delta)-n\delta-\gamma\\ & =0. \end{align*}

Hence, by the property of the infinitesimal generator, Eq. (4.1) is a martingale for the process $X(t)$.

Theorem 2. Let $\tau _{0}$ and $\tau _{1}$ be the stopping times that end the server inactivity period (when the process stays in $\mathbb {E}_{0}$) and the server occupation period (when the process stays in $\mathbb {E}_{1}$), respectively. The processes

(4.10)\begin{align} & f(C(t),N(t),Y(t),\mathbf{R}(t),t)-f(0,N(0),0,\mathbf{R}(0),0)\nonumber\\ & \quad -\int_{0}^{t}\mathcal{G}f(0,N(s),0,\mathbf{R}(s),s)\,ds \quad \text{for} \ t\in [0,\tau_{0}] \end{align}

and

(4.11)\begin{align} & f(C(t),N(t),Y(t),\mathbf{R}(t),t)-f(1,N(0),Y(0),\mathbf{R}(0),0)\nonumber\\ & \quad -\int_{0}^{t}\mathcal{G}f(1,N(s),Y(s),\mathbf{R}(s),s)\,ds \quad \text{for} \ t\in [0,\tau_{1}] \end{align}

are martingales for any function $f \in \mathfrak {D}(\mathcal {G})$.

Proof. Define the following process

$$M^{f}(t)=f(X(t))-f(X(0))-\int_{0}^{t}\mathcal{G}f(X(s))\,ds$$

where $\mathcal {G}$ is the infinitesimal generator of the PDMP $X(t)$ and $f$ is a measurable function satisfying (i)–(iii) in Theorem 5.5 of Davis [Reference Davis7]. Hence, according to Proposition 14.13 in Davis [Reference Davis8], $M^{f}(t)$ is a martingale.

Using the generators Eqs. (4.3) and (4.4), the result follows immediately.

Note that according to the treatment of our model as a PDMP, stopping times $\tau _{0}$ and $\tau _{1}$ are given by the random variables $\min (A(t)$, $R_{1}(t))$ and $Y(t)$, respectively. The component $A(t)$ refers to the inter-arrival time which is exponentially distributed with parameter $\lambda$.

5. Mean number of customers in the orbit

In this section, we will derive the expected number of customers blocked in the orbit during the periods of inactivity and occupation of the server separately. To this end, we will mainly use the result of Theorem 2. In fact, authors in Dassios and Zhao [Reference Dassios and Zhao6] have shown that this method based on martingales allows to calculate any moment of $N(t)$ without taking into account the stability condition (see Theorems 3.6 and 3.8).

Theorem 3. The conditional expectation of the number of blocked customers $N(t)$ given $N(0)=n_{0}$ and $Y(0)=y_{0}$ (when $Y(t)\in \mathbb {E}_{1}\cup \partial ^{*}\mathbb {E}_{1}$) is given by:

$$E[N(t)|N(0)=n_{0}]= \begin{cases} n_{0}+ \lambda t & \text{for} \ t \in [0,\tau_{0}];\\ n_{0}+\lambda t+\dfrac{1}{\mu_{1}} (y_{0}-t)-1 & \text{for}\ t \in [0,\tau_{1}]. \end{cases}$$

where $\mu _{1}=\int _{0}^{\infty }y\,dF(y)$.

Proof. By setting $f(i,n,y,r_{1},\ldots,r_{n},t)=y+n\mu _{1}$ and verifying the conditions Eqs. (4.5)–(4.8), we have

$$\mathcal{G}f(0,n,0,r_{1},r_{2},\ldots,r_{n},t)=\lambda \mu_{1} \quad \text{and} \quad \mathcal{G}f(1,n,y,r_{1},r_{2},\ldots,r_{n},t)=\lambda \mu_{1}-1 .$$

According to Theorem 2, $Y(t)+N(t)\mu _{1}-n_{0}\mu _{1}-\int _{0}^{t}\lambda \mu _{1}\,ds$ and $Y(t)+N(t)\mu _{1}-y_{0}-n_{0}\mu _{1}-\int _{0}^{t}(\lambda \mu _{1}-1)\,ds$ are martingales. Hence, for $t \in [0,\tau _{0}]$

(5.1)\begin{equation} E[N(t)|N(0)=n_{0}]= \dfrac{1}{\mu_{1}} E[Y(t)]+\lambda t+n_{0} \end{equation}

and for $t \in [0,\tau _{1}]$

(5.2)\begin{equation} E[N(t)|N(0)=n_{0},Y(0)=y_{0}]=n_{0}+\lambda t+ \dfrac{1}{\mu_{1}} (y_{0}-t-E[Y(t)]). \end{equation}

Besides, we have

$$E[Y(t)]=\begin{cases} 0 & \text{for}\ t\in [0,\tau_{0}];\\ \mu_{1} & \text{for}\ t\in [0,\tau_{1}]. \end{cases}$$

By substituting $E[Y(t)]$ in Eqs. (5.1) and (5.2), the result follows.