Fluid limits of many-server retrial queues with nonpersistent customers

Abstract

This work considers a many-server retrial queueing system in which nonpersistent (impatient) customers with i.i.d., generally distributed service times and independent sequences of i.i.d., generally distributed inter-attempt times. A newly arrived customer attempts to obtain service immediately upon arrival and joins a retrial orbit with probability $p\in [0,1]$ if all servers are busy, and re-attempts to obtain service after a random amount of time until it gets service. The dynamics of the system is represented in terms of a family of measure-valued processes that keep track of the amounts of time that each customer being served has been in service and the waiting times of customers in the retrial orbit since their previous attempts to obtain service. Under some mild assumptions, as both the arrival rate and the number of servers go to infinity, a law of large numbers (or fluid) limit is established for this family of processes with the aid of the one-dimensional Skorokhod map and a contraction map. The limit is shown to be the unique solution to the so-called (extended) fluid model equations. In addition, the set of invariant states for the (extended) fluid model equations is established and is used to yield some steady state performance measures, such as the steady state blocking probability and the steady state number of customers in the retrial orbit.

Appendix A: Proof of Lemma 4

Fix $T\in [0,\infty )$ and a bounded measurable function $\psi$ on $[0,H^a)\times {\mathbb R}_+$. It follows from the definition of $w^{(N)}_{ij}$ in Sect. 2.2 that for $0\le s<t<\infty$, $m\in {\mathbb N}$ and $i \in {\mathbb Z}$, the quantity

$$\begin{aligned} \sum _{j=m}^\infty \sum _{u\in (s,t]} {1\!\!1}_{\left\{ \frac{{{d}}w^{(N)}_{ij}}{{{d}}t}(u-) >0,\ \frac{{{d}}w^{(N)}_{ij} }{{{d}}t}(u+)=0\right\} } \end{aligned}$$

represents the number of attempts that customer $i$ (if such a customer exists) has made in $(s,t]$ after its $m$th attempt and before receiving service. We first consider those customers who arrived before time $0$. For each customer $i$ in the retrial orbit at time $0$, let $j^{(N)}(i)$ be the number of failed attempts it has made by time $0$ and for each customer $i$ not in the retrial orbit, let $j^{(N)}(i)=\infty$. Thus, ${{d}}w^{(N)}_{ij^{(N)}(i)}/{{d}}t(0+)>0$ if $j^{(N)}(i)<\infty$. Let $\kappa _i^{(N)}$ be the time since time $0$ that customer $i$ needs to wait till the next attempt. Let $M_i^{(N)}$ be the (delayed) renewal process defined by

$$\begin{aligned} M_i^{(N)}(t) = \max \left\{ n\ge 0:\ \kappa _i^{(N)}+\sum _{l=1}^n u^i_{j^{(N)}(i)+l}\le t\right\} ,\qquad t\in [0,\infty ) \end{aligned}$$

(4.28)

and

$$\begin{aligned} U_i^{(N)}(t) = \max \left\{ n\ge 0:\ \sum _{l=1}^n u^i_{j^{(N)}(i)+l}\le t\right\} ,\qquad t\in [0,\infty ). \end{aligned}$$

(4.29)

Obviously, for $t\in [0,\infty )$,

$$\begin{aligned} M_i^{(N)}(t)=U_i^{(N)}((t-\kappa _i^{(N)})^+). \end{aligned}$$

(4.30)

Observe that for $0\le s<t<\infty$,

$$\begin{aligned} M_i^{(N)}(t)-M_i^{(N)}(s)&= U_i^{(N)}((t-\kappa _i^{(N)})^+)-U_i^{(N)}((s-\kappa _i^{(N)})^+)\\&\le \sup _{0\le l \le t} (U_i^{(N)}(l+(t-s))-U_i^{(N)}(l)) \end{aligned}$$

and

$$\begin{aligned} {1\!\!1}_{\{M_i^{(N)}(t)+j^{(N)}(i)\ge m\}}={1\!\!1}_{\{U_i^{(N)}((t-\kappa _i^{(N)})^+)+j^{(N)}(i)\ge m\}}\le {1\!\!1}_{\{U_i^{(N)}(t)+j^{(N)}(i)\ge m\}}. \end{aligned}$$

Then, it is clear that

$$\begin{aligned}\begin{array}{l} {{d}}s\sum _{j=m}^\infty \sum _{u\in (s,t]} {1\!\!1}_{\left\{ \frac{{{d}}w^{(N)}_{ij}}{{{d}}t}(u-) >0,\ \frac{{{d}}w^{(N)}_{ij} }{{{d}}t}(u+)=0\right\} } \\ \qquad {{d}}s \le {1\!\!1}_{\{M_i^{(N)}(t)+j^{(N)}(i)\ge m\}}(M_i^{(N)}(t)-M_i^{(N)}(s))\\ \qquad \le {1\!\!1}_{\{U_i^{(N)}(t)+j^{(N)}(i)\ge m\}}\sup _{0\le l \le t} (U_i^{(N)}(l+(t-s))-U_i^{(N)}(l)). \end{array} \end{aligned}$$

(4.31)

We next consider those customers arriving after time zero. For $i=1,\cdots ,E^{(N)}(t)$, let $M_i^{(N)}$ be the (delayed) renewal process defined by

$$\begin{aligned} M_i^{(N)}(t) = \max \left\{ n\ge 0:\ \zeta _i^{(N)}+\sum _{l=1}^n u^i_{l}\le t\right\} ,\qquad t\in [0,\infty ) \end{aligned}$$

(4.33)

and

$$\begin{aligned} U_i^{(N)}(t) = \max \left\{ n\ge 0:\ \sum _{l=1}^n u^i_{l}\le t\right\} ,\qquad t\in [0,\infty ). \end{aligned}$$

(4.34)

Thus, in this case, for $t\in [0,\infty )$,

$$\begin{aligned} M_i^{(N)}(t)=U_i^{(N)}((t-\zeta _i^{(N)})^+). \end{aligned}$$

(4.35)

Then, an similar argument in deriving (4.31) yields that

$$\begin{aligned}&\sum _{j=m}^\infty \sum _{u\in (s,t]} {1\!\!1}_{\left\{ \frac{{{d}}w^{(N)}_{ij}}{{{d}}t}(u-) >0,\ \frac{{{d}}w^{(N)}_{ij} }{{{d}}t}(u+)=0\right\} } \\&\quad \le {1\!\!1}_{\{U_i^{(N)}(t)\ge m-1\}}\sup _{0\le l \le t} (U_i^{(N)}(l+(t-s))-U_i^{(N)}(l)). \nonumber \end{aligned}$$

(4.36)

Notice that $\{\sup _{0\le l \le t} (U_i^{(N)}(l+(t-s))-U_i^{(N)}(l)), i\in {\mathbb Z}\}$ is a sequence of i.i.d. random variables (independent of $N$) and is independent of $\mathcal {E}^{(N)}_0$ and $E^{(N)}$ due to the assumption on $\{\{u^i_j, j\in {\mathbb N}\},i\in {\mathbb Z}\}$ and

$$\begin{aligned} \mathbb {E}\left[ \sup _{0\le l \le t} (U_1^{(N)}(l+(t-s))-U_1^{(N)}(l))\right] \le 2\mathbb {E}[U_1^{(N)}(2t)]<\infty . \end{aligned}$$

Combining (4.9) and (4.36) and (4.31) with $s=0$, we have

$$\begin{aligned} \begin{array}{l} {{d}}s \mathbb {E}\left[ \sup _{0\le s\le T}\left| \sum _{j=m}^\infty S^{(N),j}_\psi (s) \right| \right] \\ \qquad {{d}}s \le ||\psi ||_{\infty } \mathbb {E}\left[ \sum _{i=-\mathcal {E}^{(N)}_0+1}^{0} {1\!\!1}_{\{j^{(N)}(i)<\infty , U_i^{(N)}(T)+j^{(N)}(i)\ge m\}}\sup _{0\le l \le T} (U_i^{(N)}(l+T)-U_i^{(N)}(l))\right] \\ \qquad \qquad \qquad {{d}}s + ||\psi ||_{\infty }\mathbb {E}\left[ \sum _{i=1}^{E^{(N)}(T)} {1\!\!1}_{\{U_i^{(N)}(T)\ge m-1\}}\sup _{0\le l \le T} (U_i^{(N)}(l+T)-U_i^{(N)}(l))\right] \\ \qquad {{d}}s \le ||\psi ||_{\infty } \mathbb {E}\left[ \sup _{0\le l \le T} (U_1^{(N)}(l+T)-U_1^{(N)}(l))\right] \mathbb {E}\left[ \langle \mathbf{1}, \eta ^{(N)}_0 \rangle + E^{(N)}(T)\right] . \end{array}\qquad \qquad \end{aligned}$$

(4.37)

Thus, the first part of the lemma follows from (4.37), (4.1). The second part of the lemma, (4.11), follows from the first part of the lemma, the first inequality in (4.37) and an application of the dominated convergence theorem.