由於雲端計算對於時間和能源的高效率性、資料進接的靈活性、簡單容易使用以及維護的容易性，所以對於雲端計算的需求上升。明顯地，當需求上升時，雲端計算中心的伺服器數目也必須增加。而新購買的伺服器的計算能力可能會不同於現有的伺服器。因此，雲端計算中心可能具有異質的伺服器，也就是說每個伺服器的計算能力可能不同，所以一個使用者在不同的伺服器可能會有不同的服務時間。在這篇研究中，我們考慮三種不同的情境:具有一個伺服器門檻的M⁄Mi⁄C⁄K、具有多個伺服器門檻的M⁄Mi⁄C⁄K以及具有一個伺服器門檻和兩個壅塞門檻的M⁄Mi⁄C⁄K。我們推導所考慮系統的解析模型。我們利用疊代演算法來求得穩態機率分佈和計算感興趣的效能指標。我們感興趣的效能指標是平均系統數目、平均佇列長度，遺失機率、成功送達率、平均系統延遲與平均佇列延遲。我們研究各種系統參數對於不同系統效能的影響，例如平均抵達速率、平均服務速率、伺服器門檻、與壅塞門檻。我們也呈現傳統M⁄Mi/C⁄K 模型和M⁄Mi/C⁄K的上下限模型的效能做為比較。我們發現在適當的系統參數下，具有一個伺服器門檻的M⁄Mi⁄C⁄K會有較好的平均系統延遲相對於傳統的M⁄Mi⁄C⁄K。最後，我們自行撰寫電腦模擬程式來驗證解析結果的正確性。

The requirement for cloud computing is increasing due to its high efficiency in time and energy, flexibility in data access, simple and easy use, and ease of maintenance. Obviously, when the demand increases, the number of servers in a cloud computing center must increase. The computing power of a newly bought server may be very different from those of the existing servers. That is, the cloud computing centers may have heterogeneous servers, where the computing power of each server may be different, and thus the service time of one job at different servers may be different. In this work, we study how to minimize the average system delay in cloud computing center with heterogeneous servers, where each server may have a different average service rate. We consider three different scenarios: M⁄Mi⁄C⁄K with single server threshold, M⁄Mi⁄C⁄K with multiple server thresholds, and M⁄Mi⁄C⁄K with single server threshold and double congestion thresholds. The analytical models are derived for the systems considered. We develop an iterative algorithm to find the steady state probability distribution and the performance measures of interest are computed. The performance measures of interest are average number in system, average queue length, loss probability, throughput, average system delay, and average queueing delay. The effect of various system parameters on different performance measures are studied, e.g., the mean arrival rate, the mean service rate, the server threshold, and congestion threshold. For comparison, we present the performance of not only the traditional M⁄Mi⁄C⁄K but also the lower (upper) bound models. It is shown that M⁄Mi⁄C⁄K with single server threshold outperforms M⁄Mi⁄C⁄K in terms of the average system delay with the appropriate system paremeters. Finally, the computer simulation is written to verify the accuracy of the analytical results.

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