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# The (functional) law of the iterated logarithm of the sojourn time for a multiclass queue

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• Two types of the law of iterated logarithm (LIL) and one functional LIL (FLIL) are established for the sojourn time process for a multiclass queueing model, having a priority service discipline, one server and $K$ customer classes, with each class characterized by a batch renewal arrival process and independent and identically distributed (i.i.d.) service times. The LIL and FLIL limits quantify the magnitude of asymptotic stochastic fluctuations of the sojourn time process compensated by its deterministic fluid limits in two forms: the numerical and functional. The LIL and FLIL limits are established in three cases: underloaded, critically loaded and overloaded, defined by the traffic intensity. We prove the results by a approach based on strong approximation, which approximates discrete performance processes with reflected Brownian motions. We conduct numerical examples to provide insights on these LIL results.

Mathematics Subject Classification: Primary: 60K25, 90B36; Secondary: 90B22, 68M20.

 Citation: • • Figure 1.  The LIL limits in Example 3

Table 1.  The LIL and FLIL limits for (Ⅰ) in Example 1

 $k$ 1 2 3 4 5 6 $Z^*_k=Z^*_{sup, k}$ $0$ $0$ $0$ $\sqrt{3}$ $\sqrt{3.9}$ $\sqrt{4.8}$ $Z^*_{inf, k}$ $0$ $0$ $0$ $0$ $-\sqrt{3.9}$ $-\sqrt{4.8}$ $\mathcal{K}_{Z_{k}}$ $\{0\}$ $\{0\}$ $\{0\}$ $\Phi(\mathcal{G} (\sqrt{3}))$ $\mathcal{G} (\sqrt{3.9})$ $\mathcal{G} (\sqrt{4.8})$ $\mathcal{S}^*_{k}=\mathcal{S}^*_{sup, k}$ $0$ $0$ $0$ $10\sqrt{3}$ $\mathcal{S}^*_{inf, k}$ $0$ $0$ $0$ $0$ $\mathcal{K}_{\mathcal{S}_{k}}$ $\{0\}$ $\{0\}$ $\{0\}$ $\Phi(\mathcal{G} (10\sqrt{3}))$

Table 2.  The LIL and FLIL limits for (Ⅱ) in Example 1

 $k$ 1 2 3 4 5 6 $Z^*_k=Z^*_{sup, k}$ $0$ $0$ $0$ $\sqrt{3.6}$ $\sqrt{4.5}$ $\sqrt{5.4}$ $Z^*_{inf, k}$ $0$ $0$ $0$ $-\sqrt{3.6}$ $-\sqrt{4.5}$ $-\sqrt{5.4}$ $\mathcal{K}_{Z_{k}}$ $\{0\}$ $\{0\}$ $\{0\}$ $\mathcal{G} (3.6)$ $\mathcal{G} (4.5)$ $\mathcal{G} (5.4)$ $\mathcal{S}^*_{k}=\mathcal{S}^*_{sup, k}$ $0$ $0$ $0$ $20\sqrt{3}$ $\mathcal{S}^*_{inf, k}$ $0$ $0$ $0$ $-20\sqrt{3}$ $\mathcal{K}_{\mathcal{S}_{k}}$ $\{0\}$ $\{0\}$ $\{0\}$ $\mathcal{G} (20\sqrt{3})$

Table 3.  The LIL and FLIL limits for (Ⅱ) in Example 2

 $k$ 1 2 3 4 5 $Z^*_k=Z^*_{sup, k}$ $0$ $0$ $C_{3}$ $C_{4}$ $C_{5}$ $Z^*_{inf, k}$ $0$ $0$ $0$ $-C_{4}$ $-C_{5}$ $\mathcal{K}_{Z_{k}}$ $\{0\}$ $\{0\}$ $\Phi(\mathcal{G} (C_{3}))$ $\mathcal{G} (C_{4})$ $\mathcal{G} (C_{5})$ $\mathcal{S}^*_{k}=\mathcal{S}^*_{sup, k}$ $0$ $0$ $5C_{3}$ $\mathcal{S}^*_{inf, k}$ $0$ $0$ $0$ $\mathcal{K}_{\mathcal{S}_{k}}$ $\{0\}$ $\{0\}$ $\Phi(\mathcal{G} (5C_{3}))$

Table 4.  The LIL and FLIL limits for (Ⅲ) in Example 2

 $k$ 1 2 3 4 5 $Z^*_{k}=Z^*_{sup, k}$ $0$ $0$ $D_{3}$ $D_{4}$ $D_{5}$ $Z^*_{inf, k}$ $0$ $0$ $-D_{3}$ $-D_{4}$ $-D_{5}$ $\mathcal{K}_{Z_{k}}$ $0$ $0$ $\mathcal{G}(D_{3})$ $\mathcal{G}(D_{4})$ $\mathcal{G}(D_{5})$ $\mathcal{S}^*_{k}=\mathcal{S}^*_{sup, k}$ 0 0 $D$ $\mathcal{S}^*_{inf, k}$ $0$ $0$ $-D$ $\mathcal{K}_{\mathcal{S}_{k}}$ $0$ $0$ $\mathcal{G}(D)$
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