{
  "id": "1208.5552",
  "version": "v2",
  "published": "2012-08-28T04:02:06.000Z",
  "updated": "2014-08-25T17:23:39.000Z",
  "title": "A Unified Approach to Diffusion Analysis of Queues with General Patience-time Distributions",
  "authors": [
    "Junfei Huang",
    "Hanqin Zhang",
    "Jiheng Zhang"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We propose a unified approach to establishing diffusion approximations for queues with impatient customers within a general framework of scaling customer patience time. The approach consists of two steps. The first step is to show that the diffusion-scaled abandonment process is asymptotically close to a function of the diffusion-scaled queue length process under appropriate conditions. The second step is to construct a continuous mapping not only to characterize the system dynamics using the system primitives, but also to help verify the conditions needed in the first step. The diffusion approximations can then be obtained by applying the continuous mapping theorem. The approach has two advantages: (i) it provides a unified procedure to establish the diffusion approximations regardless of the structure of the queueing model or the type of patience time scaling; (ii) and it makes the diffusion analysis of queues with customer abandonment essentially the same as the diffusion analysis of queues without customer abandonment. We demonstrate the application of the approach via the $G/GI/1+GI$ system with Markov-modulated service speeds in the traditional heavy-traffic regime and the $G/GI/n+GI$ system in the Halfin-Whitt regime.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-08-28T04:02:06.000Z",
      "title": "Diffusion Approximations for Queueing Systems with Customer Abandonment",
      "abstract": "This paper studies the diffusion approximations for queueing systems with customer abandonment. As queueing systems arise from various applications and the abandonment phenomenon has been recognized to be significant, we develop the diffusion analysis for such systems within a general framework for modeling the abandonment. In light of the recent work by Dai and He (2010), our first step is to show that, under an appropriate condition, the diffusion-scaled cumulative number of customers who have abandoned is asymptotically close to a function of the diffusion-scaled queue length process. Applying this asymptotic relationship, the diffusion approximations for such systems are established in a wide range of heavy traffic regimes, including the traditional heavy traffic regime, the non-degenerate slowdown regime and the Halfin-Whitt regime. The key step is to develop mappings with nice properties that can characterize the system dynamics by connecting the primitive processes such as the arrival, service and abandonment processes to the processes underlying the systems such as the queue length or the total head count.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-08-25T17:23:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "diffusion approximations",
      "queueing systems",
      "customer abandonment",
      "traditional heavy traffic regime",
      "diffusion-scaled queue length process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.5552H"
    }
  }
}