The recent growth of the Internet use overloads networking systems and degrades the quality of communications, e.g., bandwidth loss, packet drops, delay of responses, etc. To overcome such degradation of the communication quality, the notion of Quality of Service (QoS) has received attention in practice. In general, QoS switches have several queues and each queue has several slots to store arriving packets. Since network traffic changes frequently, QoS switches need to control arriving packets to maximize the total priorities of transmitted packets, where the priorities are given by nonnegative values and correspond to the quality of service required to each packet. In this paper, we derive lower bounds for the competitive ratio of deterministic multi-queue nonpreemptive QoS problem of priorities 1 and a > 1, and show that for any α ≥ 1, 1 + /αln(α/(α-1)) if α≥α{sup}*; 1/(1-e{sup}(-το)) if 1≤α<α{sup}*, where a{sup}*≈1.657 and τ{sub}0 is a root of the equality that e{sup}(-τ)(1/α+τ)=1-e{sup}τ, As an immediate result, this shows a lower bound 1.466 for the competitive ratio of deterministic multi-queue nonpreemptive QoS problem of single priority, which slightly improves the best known lower bound 1.366.
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