Continuity of a queueing integral representation in the ${M}_{mathbf{1}}$ topology
Continuity of a queueing integral representation in the ${M}_{mathbf{1}}$ topology
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${M}_{mathbf{1}}$ 拓扑中排队积分表示的连续性
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
W. Whitt
中科院分区：
文献类型：
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作者：
G. Pang;W. Whitt
当函数空间D具有Skorohod M1拓扑时，建立了函数x到函数y的积分表示y（t）=x（t）+int 0 ^th（y（s））ds，tge 0 $的连续性.我们应用这种积分表示与连续映射定理，以建立大流量的随机过程的限制多服务器排队模型时，极限过程有不匹配的跳跃收敛过程中，可以发生突发性的到达过程或服务中断。M1-连续性的证明是基于M1收敛性的一个新的刻画，其中参数表示的时间部分关于Lebesgue测度是绝对连续的，导数是一致有界的，并且收敛于L1.
We establish continuity of the integral representation $y(t)=x(t)+int_0^th(y(s)) ds$, $tge0$, mapping a function $x$ into a function $y$ when the underlying function space $D$ is endowed with the Skorohod $M_1$ topology. We apply this integral representation with the continuous mapping theorem to establish heavy-traffic stochastic-process limits for many-server queueing models when the limit process has jumps unmatched in the converging processes as can occur with bursty arrival processes or service interruptions. The proof of $M_1$-continuity is based on a new characterization of the $M_1$ convergence, in which the time portions of the parametric representations are absolutely continuous with respect to Lebesgue measure, and the derivatives are uniformly bounded and converge in $L_1$.