General hazard-type scaling of abandonment time distribution for a G/Ph/n+GI queue in the Halfin–Whitt heavy-traffic regime

Abstract

We consider the diffusion approximation of a G/Ph/n queue with customer abandonment in the Halfin–Whitt heavy-traffic regime and extend the conventional locally Lipschitz hazard-type scaling of abandonment time distribution to a more general scaling in the study. Under that new scaling of abandonment, not only the non-locally Lipschitz hazard-type case such as the non-locally bounded hazard-rate scaling which has never been solved to date, but also a wider range of abandonment time distributions including the non-absolutely continuous ones becomes subject to the analysis of diffusion approximation. Due to the general character of our scaling scheme of abandonment time, the stochastic equation for the limit of C-tight, scaled, and centered customer-count processes contains a nonlinear drift term as the limit of abandonment-count process, which does not satisfy the local Lipschitz condition so that the continuous mapping method in the literature of this area is invalid in our case. Instead, applying the Girsanov transformation to the localized multidimensional equation satisfied by the limit of customer-count process in multiphase of service time, we show the uniqueness in law of the solution to establish the desired diffusion approximation.

Appendix

In this appendix, we put the following technical lemma that is used in the identification of the limit for the scaled abandonment-count processes, i.e., Lemma 2.7. It is a minor modification of Lemma 3.1 in Yamada [24] and also a generalization of Lemma 8.3 in Dai and Dai [4].

Lemma 4.1

Suppose that $y^n(s), s\ge 0, n\in {{\mathcal {N}}},$ is a sequence of nondecreasing, right-continuous functions such that $y^n(0)=0, n\in {{\mathcal {N}}},$ and $y^n$ converges to $y$ in ${\mathbb {D}}([0, \infty ), {{\mathcal {R}}}^1)$ as $n$ tends to infinity, where $y$ is a continuous function. Also assume that a sequence of functions $f^n(s), s\ge 0, n\in {{\mathcal {N}}},$ is such that for each $M>0$,

$$\begin{aligned} \sup _{n\in {{\mathcal {N}}}} \sup _{0\le s \le M} |f^n(s)|<\infty \quad \text {and} \quad \sup _{0\le s \le M} |f(s)|<\infty , \end{aligned}$$

and for $y(\mathrm{d}s)$-a.e. $s$,

$$\begin{aligned} f^n(s)\longrightarrow f(s), \end{aligned}$$

as $n$ tends to infinity. Moreover, suppose that there exists a sequence of step functions $g_k(s), s\ge 0, k\in {{\mathcal {N}}},$ such that for each $s\ge 0$,

$$\begin{aligned} g_k(s)\longrightarrow f(s) \end{aligned}$$

as $k$ tends to infinity. Then we have

$$\begin{aligned} \int ^t_0 f^n(s)y^n(\mathrm{d}s)\longrightarrow \int ^t_0 f(s)y(\mathrm{d}s) \end{aligned}$$

as $n$ tends to infinity, uniformly for $t$ in any compact subset of ${{\mathcal {R}}}^1_+$.

$($ A function $g(s), s\ge 0,$ is said to be a step function if $g(s)=c_i, s_{i-1}<s<s_{i},$ for some subdivision $\{s_i \}$ of $[0, \infty ))$.