隨著科技的發展，物聯網更是廣泛利用在生活中。物聯網時常應用在工業、交通、和行動通訊系統像是5G。一方面，傳統的點對點流量增長導致頻寬不足和壅塞。另一方面，長的端到端距離降低了服務品質。因此，如何實現良好的端到端的機器對機器的通訊變成是很重要的課題。人們提出一種稱為發布/訂閱系統 (pub/sub system) 的架構。發布/訂閱系統由發布器、中介器(broker)和訂閱器組成。它可以讓機器對機器(發布器與訂閱器)在空間與時間上進行去耦合。在發布/訂閱系統中，發布器到中介器的距離和中介器到訂閱器的距離是時變的。具有兩種服務狀態，也就是，ON和OFF的間歇性排隊模型被用來對上述情況進行建模。不同的服務狀態有不同的服務速率。在我們的研究中，我們提出擁有三種服務狀態，也就是，ON、Mid和OFF，的間歇性排隊模型來進一步改善通訊服務品質。中介器由一個M/M/1/K佇列和一個間歇性佇列串聯而成。在中間件層的整個發布/訂閱系統由兩個M/M/1/K佇列和兩個間歇性佇列串聯而成。除此之外，為了表示事件的緊急性，發布的事件具有生存期限。當事件在佇列裡過期時，會立即離開系統。再者，當事件進入不同的佇列，它會重新產生生存期限。首先，我們推導所考慮系統的解析模型。接下來，我們使用疊代演算法來找到穩態機率分佈和感興趣的效能指標。其次，我們撰寫了模擬程式來證明解析結果的準確性。第三，我們研究了系統參數，例如到達速率，對感興趣的效能指標的影響。最後，我們比較具有兩個服務狀態的間歇性佇列和具有三個服務狀態的間歇性佇列的感興趣效能指標。

With the development of science and technology, the Internet of Things is widely used in life. The Internet of Things is often used in industry, transportation, and mobile communication systems such as 5G. On one hand, the traditional point-to-point traffic growth has led to bandwidth shortage and congestion. On the other hand, the long end-to-end distance degrades the quality of service. Therefore, how to achieve good end-to-end machine-to-machine communication has become a very important topic. People proposed an architecture called pub/sub system. The pub/sub system consists of publishers, brokers, and subscribers. It allows machine-to-machine (publisher and subscriber) to decouple in space and time. In a pub/sub system, the distance from publisher to broker and that from broker to subscriber is time-varying. The intermittent queuing model with two service states, i.e., ON and OFF, has been used to model the above situation. A different service state has a different service rate. In our research, we propose an intermittent queuing model with three service states, i.e., ON, Mid, and OFF to further improve the communication quality of service. The broker is composed of an M/M/1/K queue and an intermittent queue in series. The entire pub/sub system in the middleware layer is composed of two M/M/1/K queues and two intermittent queues in series. In addition, to indicate the emergency of the events, the published event has a lifetime limit. When the event expires in the queue, it will immediately leave the system. Furthermore, when the event enters a different queue, it will regenerate the lifetime limit. First, we derive the analytical models for the considered system. Next, an iterative algorithm is used for finding the steady state probability distribution and the interested performance measures. Second, we write simulation programs to verify the accuracy of the analytical results. Third, we study the influence of system parameters, e.g., the arrival rate, on interested performance measures. Finally, we compare the interested performance measures for the intermittent queue with two service states and that with three states.

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