Optimal processing rate and buffer size of a jump-diffusion processing system

Abstract

In this paper, we propose a reflected jump-diffusion model for processing systems with finite buffer size. We derive an analytic expression for the total expected discounted managing cost, which facilitates finding (numerically) the optimal processing rate and buffer size that minimize the total cost. Moreover, the formula for steady-state density of the processing system is obtained.

Appendices

Appendix A: Proof of Theorem 3

Define

$$\begin{aligned} g_0(x):= \begin{cases} 1,& x\geq b,\\ \sum_{i=1}^4C_{0,i}(\beta;\mu,b)e^{-{\delta}_{i,\beta} (b-x)},&0\leq x<b,\\ g_0(0),&x<0, \end{cases} \end{aligned}$$

(A.1)

where C _0,i (β;μ,b)’s satisfy the system of linear equations (3.7). From the last equation of (3.7), we have g ₀(⋅) is continuous at b. By doing the integration in three regions (−∞,−x], (−x,b−x) and [b−x,∞), we get

(A.2)

where ϕ(δ)=G(δ)−β, and

(A.3)

is the infinitesimal generator of the process Z. Recall from (3.7) that

$$\begin{aligned} \sum_{i=1}^4C_{0,i}(\beta; \mu,b)\frac{\gamma_1}{\gamma_1-{\delta }_{i,\beta}}=1,\qquad \sum_{i=1}^4C_{0,i}( \beta;\mu,b)\frac{{\delta}_{i,\beta}}{\gamma _2+{\delta}_{i,\beta}}e^{-{\delta}_{i,\beta} b}=0. \end{aligned}$$

We have

(A.4)

Note that, from the second last equation in (3.7), $g_{0}'(0)=0$. Now by Itô’s formula for jump processes (see, e.g., Protter 2004, Chapter II, Theorem 32) and the boundedness of the function g ₀(⋅), we have⁴

$$\begin{aligned} \bigl\{ M_t:=e^{-\beta(t\wedge\tau_b)}g_0 (X_{t\wedge\tau _b} ), t\geq0 \bigr\} \end{aligned}$$

(A.5)

is a martingale. By the martingale property

$$\begin{aligned} g_0(x)=\mathbb {E}_x \bigl[e^{-\beta(t\wedge\tau_b)}g_0 (X_{t\wedge \tau_b} ) \bigr] \quad\mbox{for each } t\geq0. \end{aligned}$$

(A.6)

Note that $g_{0}(X_{\tau_{b}})=1$. By the Lebesgue dominated convergence theorem, we have

$$\begin{aligned} g_0(x)=\lim_{t\rightarrow\infty} \mathbb {E}_x \bigl[e^{-\beta(t\wedge\tau _b)}g_0(X_{t\wedge\tau_b}) \bigr] = \mathbb {E}_x \bigl[e^{-\beta\tau_b}g_0(X_{\tau_b}) \bigr]= \mathbb {E}_x \bigl[e^{-\beta\tau_b} \bigr]. \end{aligned}$$

(A.7)

We have proved (3.4).

To prove (3.5), define

$$\begin{aligned} g_1(x):= \begin{cases} 0, & x \geq b,\\ \sum_{i=1}^4C_{1,i}(\beta;\mu,b)e^{-{\delta}_{i,\beta} (b-x)},&0\leq x<b,\\ g_1(0)-x,& x<0, \end{cases} \end{aligned}$$

(A.8)

where C _1,i (β;μ,b)’s satisfy the system of linear equations (3.8). By doing the integration in four regions (−∞,−x), [−x,b−x), [b−x,+∞) and recalling the first two equations in (3.8), we have

(A.9)

Now by Itô’s formula for jump processes, we have

(A.10)

Here the summation is over all jump locations before time τ _b . Although the function g ₁(⋅) is not bounded, its boundedness on [0,b] as well as the compound Poisson jumps with exponentially distributed jump size guarantee the validness of the above equation. Note that $g_{1}(X_{\tau_{b}})=0$, $g_{1}'(0)=-1$ (from the second last equation in (3.8)), and g ₁(X _s )−g ₁(X _s−+ΔZ _s )=−ΔL _s . We have

$$\begin{aligned} g_1(x)= \mathbb {E}_x \biggl[\int_0^{\tau_b}e^{-\beta s} {\mathrm {d}}L^c_s + \sum_{0<s\leq\tau_b}e^{-\beta s} \Delta L_s \biggr]=\mathbb {E}_x \biggl[\int _0^{\tau_b}e^{-\beta s} {\mathrm {d}}L_s \biggr], \end{aligned}$$

(A.11)

which concludes the proof of (3.5).

Formula (3.6) can be proved in a similar way by defining

$$\begin{aligned} g_2(x):= \begin{cases} 1,& x>b+y,\\ 0,& b\leq x \leq b+y,\\ e^{-\gamma_1 y}\sum_{i=1}^4C_{2,i}(\beta;\mu,b)e^{-{\delta }_{i,\beta} (b-x)},&0\leq x<b,\\ g_2(0),& x<0, \end{cases} \end{aligned}$$

(A.12)

where C _2,i (β;μ,b)’s satisfy the system of linear equations (3.9). The proof of the theorem is completed.

Appendix B: Proof of Theorem 5

Define

$$\begin{aligned} g_3(x):= \begin{cases} g_3(b),& x\geq b,\\ \sum_{i=1}^4C_{3,i}(\beta;\mu,b)e^{{\delta}_{i,\beta} x},&0< x<b,\\ 1,&x\leq0, \end{cases} \end{aligned}$$

(B.1)

where C _3,i (β;μ,b)’s satisfy the system of linear equations (3.14). Similarly, we can show that 𝒜g ₃(x)−βg ₃(x)=0,0≤x<b. Note that $g_{3}(X_{\tau_{0}})=1$. From a martingale argument and by the Lebesgue dominated convergence theorem, we have

$$\begin{aligned} g_3(x)=\lim_{t\rightarrow\infty} \mathbb {E}_x \bigl[e^{-\beta(t\wedge\tau _0)}g_3\bigl(X(t\wedge\tau_0)\bigr) \bigr] =\mathbb {E}_x \bigl[e^{-\beta\tau_0}g_3\bigl(X( \tau_0)\bigr) \bigr]=\mathbb {E}_x \bigl[e^{-\beta\tau_0} \bigr]. \end{aligned}$$

(B.2)

We have proved (3.11). To prove (3.12), define

$$\begin{aligned} g_4(x):= \begin{cases} x-b+g_4(b), & x > b,\\ \sum_{i=1}^4C_{4,i}(\beta;\mu,b)e^{{\delta}_{i,\beta} x},&0< x\leq b,\\ 0,& x\leq0, \end{cases} \end{aligned}$$

(B.3)

where C _4,i (β;μ,b)’s satisfy the system of linear equations (3.15). Again we can show that 𝒜g ₄(x)−βg ₄(x)=0,0≤x<b. Now by Itô’s formula for jump processes (see, e.g., Protter 2004, Chapter II, Theorem 32), we have

(B.4)

Here the summation is over all jump locations before time τ ₀. Although the function g ₄(⋅) is not bounded, its boundedness on [0,b] as well as the compound Poisson jumps with exponentially distributed jump size guarantee the validness of the above equation. Note that $g_{4}(X_{\tau_{0}})=0$, $g_{4}'(b)=1$ (from the second last equation in (3.15)), and g ₄(X _s )−g ₄(X(s−)+ΔZ _s )=−ΔU _s . We have

$$\begin{aligned} g_4(x)= \mathbb {E}_x \biggl[\int_0^{\tau_0}e^{-\beta s} {\mathrm {d}}U^c_s + \sum_{0<s\leq\tau_0}e^{-\beta s} \Delta U_s \biggr]=\mathbb {E}_x \biggl[\int _0^{\tau_0}e^{-\beta s} {\mathrm {d}}U_s \biggr]. \end{aligned}$$

(B.5)

Formula (3.13) can be proved in a very similar way by defining

$$\begin{aligned} g_5(x):= \begin{cases} g_5(b),& x>b,\\ e^{-\gamma_1 y} \sum_{i=1}^4C_{5,i}(\beta;\mu,b)e^{-{\delta }_{i,\beta} (b-x)},& 0 < x \leq b,\\ 0, & -y\leq x\leq0,\\ 1, & x<-y, \end{cases} \end{aligned}$$

(B.6)

with C _5,i (β;μ,b)’s satisfying the system of linear equations (3.16). This concludes the proof of the theorem.

Appendix C: Proof of Theorem 7

Define

$$\begin{aligned} g_6(x):= \begin{cases} 1,& x\geq v,\\ \sum_{i=1}^4D_i(\beta;u,v)e^{{\delta}_{i,\beta} x},&u<x<v,\\ 0,&x\leq u, \end{cases} \end{aligned}$$

(C.1)

where D _i (β;u,v)’s satisfy the system of linear equations (3.20). By doing the integration in three regions (−∞,u−x], (u−x,v−x) and [v−x,∞) and using the first two equations in (3.20), we have

Noting that $g_{6}(Z_{\tau_{[u,v)}})=1_{\{Z_{\tau_{[u,v)}}\geq v\}}$, we have, by the Lebesgue dominated convergence theorem,

$$\begin{aligned} g_6(x)=\lim_{t\rightarrow\infty} \mathbb {E}_x \bigl[e^{-\beta(t\wedge\tau _{[u,v)})}g_6(Z_{t\wedge\tau_{[u,v)}}) \bigr] = \mathbb {E}_x \bigl[e^{-\beta\tau_{[u,v)}}g_6(Z_{\tau_{[u,v)}}) \bigr]= \mathbb {E}_x \bigl[e^{-\beta\tau_{[u,v)}} 1_{\{Z_{\tau_{[u,v)}}\geq v\}} \bigr]. \end{aligned}$$

We have proved (3.19). Formula (3.21) follows by letting β decrease to zero. The proof of the theorem is now completed.