Staffing large-scale service systems with distributional uncertainty

Abstract

This paper analyzes a staffing level problem for large-scale single-station queueing systems. The system manager operates an Erlang-C queueing system with a quality-of-service constraint on the probability that a customer is queued. However, in this model, the arrival rate is uncertain in the sense that even the arrival-rate distribution is not completely known to the manager. Rather, the manager has an estimate of the support of the arrival-rate distribution and the mean. The goal is to determine the number of servers needed to satisfy the quality-of-service constraint. Two cases are explored. First, the constraint is enforced on an overall delay probability, given the probability that different feasible arrival-rate distributions are selected. In the second case, the constraint has to be satisfied by every possible distribution. For both problems, asymptotically optimal solutions are developed based on Halfin–Whitt type scalings.

Appendix

Finding the centroid when $n=4$ (Figs. 4, 5, and 6).

Fig. 4

Projection of $\mathcal {D}$ when $\lambda _{}^{\omega _{3}} \le r<\lambda _{}^{\omega _{4}}$

Fig. 5

Projection of $\mathcal {D}$ when $\lambda _{}^{\omega _{2}} \le r<\lambda _{}^{\omega _{3}}$

Fig. 6

Projection of $\mathcal {D}$ when $\lambda _{}^{\omega _{1}} \le r<\lambda _{}^{\omega _{2}}$

Proof of Lemma 5

We prove the statement by contradiction. Suppose there exists a distribution $\mathbf {p}_{\textsc {t}}$ which is a maximizer of (21) and for which $p_{\textsc {t}}^{\omega _k}>0$ for $k\notin \{1,k^\prime ,k^\prime +1\}$. If $k\in \{2,\ldots ,k^\prime -1\}$, we define a new distribution $\tilde{\mathbf {p}}$ with

$$\begin{aligned} \tilde{p}^{\omega _1}= & {} p_{\textsc {t}}^{\omega _1}+\frac{\lambda _{}^{\omega _{k^\prime }}-\lambda _{}^{\omega _{k}}}{\lambda _{}^{\omega _{k^\prime }}-\lambda _{}^{\omega _{1}}}p_{\textsc {t}}^{\omega _k},\\ \tilde{p}^{\omega _{k^\prime }}= & {} p_{\textsc {t}}^{\omega _{k^\prime }}+\frac{\lambda _{}^{\omega _{k}}-\lambda _{}^{\omega _{1}}}{\lambda _{}^{\omega _{k^\prime }}-\lambda _{}^{\omega _{1}}}p_{\textsc {t}}^{\omega _k}, \end{aligned}$$

$\tilde{p}^{\omega _k}=0$ and equal probabilities for all other scenarios in $\mathbf {p}_{\textsc {t}}$. Since ${}^{(\lambda _{}^{\omega _{k}}-\lambda _{}^{\omega _{1}})}\!/_{(\lambda _{}^{\omega _{k^\prime }}-\lambda _{}^{\omega _{1}})}$ and $\tilde{\alpha }(\beta ,\lambda _{}^{\omega _{k^\prime }},\lambda _{}^{\omega _{k^\prime }})$ are both positive, $\tilde{\mathbf {p}}$ returns a larger value of the argument in (21) than $\mathbf {p}_{\textsc {t}}$, leading to a contradiction. If $k\in \{k^\prime +2,\ldots ,n\}$, define $\tilde{\mathbf {p}}$ as

$$\begin{aligned} \tilde{p}^{\omega _1}= & {} p_{\textsc {t}}^{\omega _1}+\frac{\lambda _{}^{\omega _{k^\prime +1}}-\lambda _{}^{\omega _{k}}}{\lambda _{}^{\omega _{k^\prime +1}}-\lambda _{}^{\omega _{1}}}p_{\textsc {t}}^{\omega _k},\\ \tilde{p}^{\omega _{k^\prime +1}}= & {} p_{\textsc {t}}^{\omega _{k^\prime +1}}+\frac{\lambda _{}^{\omega _{k}}-\lambda _{}^{\omega _{1}}}{\lambda _{}^{\omega _{k^\prime +1}}-\lambda _{}^{\omega _{1}}}p_{\textsc {t}}^{\omega _k}, \end{aligned}$$

$\tilde{p}^{\omega _k}=0$ and keep all other probabilities unchanged from $\mathbf {p}_{\textsc {t}}$. A contradiction is again obtained since ${}^{(\lambda _{}^{\omega _{k}}-\lambda _{}^{\omega _{1}})}\!/_{(\lambda _{}^{\omega _{k^\prime +1}}-\lambda _{}^{\omega _{1}})}>1$. $\square$

Proof of Lemma 7

We prove (30) first. Assume $d_m$ is the optimal solution to the maximization problem inside the limit for each $m\in \mathbb {Z}^+$. Then we can rewrite the left-hand side of (30) as

$$\begin{aligned} \lim _{m\rightarrow \infty }\left|\sum _{k=1}^{l-1}p^{\omega _{k}}(d_m)\tilde{\alpha }\left( \beta ,\lambda _{m}^{\omega _{l}},\lambda _{m}^{\omega _{k}}\right) +\sum _{k=l+1}^{n}p^{\omega _{k}}(d_m)\left( \tilde{\alpha }\left( \beta ,\lambda _{m}^{\omega _{l}},\lambda _{m}^{\omega _{k}}\right) -1\right) \right|. \end{aligned}$$

(35)

As m goes to infinity, $\tilde{\alpha }(\beta ,\lambda _{m}^{\omega _{l}},\lambda _{m}^{\omega _{k}})$ converges pointwise to 0 when $k<l$ and to 1 when $k>l$ (see Corollary 13 from Zan et al. [28]). Using this observation and the fact that $p^{\omega _{k}}(d_m)$ is uniformly bounded above by 1 for all m, we conclude that

$$\begin{aligned} \lim _{m\rightarrow \infty }\left[ \sum _{k=1}^{l-1}p^{\omega _{k}}(d_m)\tilde{\alpha }\left( \beta ,\lambda _{m}^{\omega _{l}},\lambda _{m}^{\omega _{k}}\right) +\sum _{k=l+1}^{n}p^{\omega _{k}}(d_m)\left( \tilde{\alpha }\left( \beta ,\lambda _{m}^{\omega _{l}},\lambda _{m}^{\omega _{k}}\right) -1\right) \right] =0. \end{aligned}$$

Then, since absolute value is a continuous function, the limit in (35) exists and equals 0, which establishes (30).

We now consider (31). We define the following quantities:

$$\begin{aligned} d_m^F\in \mathop {{{\mathrm{arg\,max}}}}\limits _{d\in \mathcal {D}}\left\{ \sum _{k=1}^{n}p^{\omega _{k}}(d)\tilde{\alpha }\left( \beta ,\lambda _{m}^{\omega _{l}},\lambda _{m}^{\omega _{k}}\right) \right\} \end{aligned}$$

and

$$\begin{aligned} d_m^H\in \mathop {{{\mathrm{arg\,max}}}}\limits _{d\in \mathcal {D}}\left\{ p^{\omega _{l}}(d)\tilde{\alpha }\left( \beta ,\lambda _{m}^{\omega _{l}},\lambda _{m}^{\omega _{l}}\right) +\sum _{k=l+1}^n p^{\omega _{k}}(d)\right\} . \end{aligned}$$

Then, the result in (30) indicates that, for any $\delta >0$, there exists an $\bar{m}$ such that for all $m>\bar{m}$,

$$\begin{aligned}&\left|\sum _{k=1}^{n}p^{\omega _{k}}\left( d_m^F\right) \tilde{\alpha }\left( \beta ,\lambda _{m}^{\omega _{l}},\lambda _{m}^{\omega _{k}}\right) -\left( p^{\omega _{l}}\left( d_m^H\right) \tilde{\alpha }\left( \beta ,\lambda _{m}^{\omega _{l}},\lambda _{m}^{\omega _{l}}\right) +\sum _{k=l+1}^n p^{\omega _{k}}\left( d_m^H\right) \right) \right|\nonumber \\&\quad \le \max \left\{ \sum _{k=1}^{n}p^{\omega _{k}}\left( d_m^F\right) \tilde{\alpha }\left( \beta ,\lambda _{m}^{\omega _{l}},\lambda _{m}^{\omega _{k}}\right) \right. \nonumber \\&\qquad \left. -\left( p^{\omega _{l}}\left( d_m^F\right) \tilde{\alpha }\left( \beta ,\lambda _{m}^{\omega _{l}},\lambda _{m}^{\omega _{l}}\right) +\sum _{k=l+1}^n p^{\omega _{k}}\left( d_m^F\right) \right) ,\right. \nonumber \\&\quad \left. \left( p^{\omega _{l}}\left( d_m^H\right) \tilde{\alpha }\left( \beta ,\lambda _{m}^{\omega _{l}},\lambda _{m}^{\omega _{l}}\right) {+}\sum _{k=l+1}^n p^{\omega _{k}}\left( d_m^H\right) \right) {-}\sum _{k=1}^{n}p^{\omega _{k}}\left( d_m^H\right) \tilde{\alpha }\left( \beta ,\lambda _{m}^{\omega _{l}},\lambda _{m}^{\omega _{k}}\right) \right\} \nonumber \\&\quad \le \delta . \end{aligned}$$

This establishes (31). $\square$