Abstract:
By using the Hille-Yosida theorem, Phillips theorem, and Fattorini theorem in functional analysis we prove that the /G/1 queueing model with vacation times has a unique nonnegative time-dependent solution. 1. Introduction The queueing system when the server become idle is not new. Miller [1] was the first to study such a model, where the server is unavailable during some random length of time for the M/G/1 queueing system. The M/G/1 queueing models of similar nature have also been reported by a number of authors, since Levy and Yechiali [2] included several types of generalizations of the classical M/G/1 queueing system. These generalizations are useful in model building in many real life situations such as digital communication, computer network, and production/inventory system [3–5]. At present, however, most studies are devoted to batch arrival queues with vacation because of its interdisciplinary character. Considerable efforts have been devoted to study these models by Baba [6], Lee and Srinivasan [7], Lee et al. [8, 9], Borthakur and Choudhury [10], and Choudhury [11, 12] among others. However, the recent progress of /G/1 type queueing models of this nature has been served by Chae and Lee [13] and Medhi [14]. In 2002, Choudhury [15] studied the /G/1 queueing model with vacation times. By using the supplementary variable technique [16] he established the corresponding queueing model and obtained the queue size distribution at a stationary (random) as well as a departure point of time under multiple vacation policy based on the following hypothesis. “The time-dependent solution of the model converges to a nonzero steady-state solution.” By reading the paper we find that the previous hypothesis, in fact, implies the following two hypothesis. Hypothesis 1. The model has a nonnegative time-dependent solution. Hypothesis 2. The time-dependent solution of the model converges to a nonzero steady-state solution. In this paper we investigate Hypothesis 1. By using the Hille-Yosida theorem, Phillips theorem, and Fattorini theorem we prove that the model has a unique nonnegative time-dependent solution, and therefore we obtain Hypothesis 1. According to Choudhury [15], the /G/1 queueing system with vacation times can be described by the following system of equations: where ; represents the probability that there is no customer in the system and the server is idle at time ; represents the probability that at time there are customers in the system and the server is on a vacation with elapsed vacation time of the server lying in . represents the probability that

Abstract:
A total of 62 green alga strains were isolated from the soils of Nanshan Mountain, Xinjiang. Used the blot method to characterize these metal resistance, the results indicated that XJU-3、XJU-28 and XJU-36 have resistance to 0. 1 mmol · L~(-1) Co~(2+) ;XJU-28 has resistance to 1 mmol·L~(-1) Zn~(2+) and Fe~(3+) , XJU-36 has resistance to 0. 05 mmol·L~(-1) Cu~(2+). Taxonomic evalua-tion of the three strains were investigated based on the morphology and the internal transcribed spacer (ITS) regions (including the 5. 8S). Based on morphological characteristics, the three strains were likely to Chlamydomonas. Phylogenetic reconstruction with the Neighbor-joining (NJ) method using sequences of ITS(including the 5. 8S) indicated that XJU-3 and XJU-28 are closed to Chlamydomonas zebra. XJU-36 is closed to Chlamydornonas petasua.