versions of the PH distributions of the new environment. The stationary version of the distribution for residual time has been well explained in Qi-Ming He [12] where it is named as equilibrium PH distribution. Randomly varying environment PH/PH/1 queue models with **bulk** arrival and **bulk** **service** have not been treated so far at any depth. In this paper the partitioning of the matrix is carried out in a way that the stationary probability vector exhibits a matrix geometric structure for PH/PH/1 **bulk** queues with random environment where the arrivals and **service** sizes are finite. Two models (A) and (B) on PH/PH/1 **bulk** queue systems with infinite storage space for customers are studied using the block partitioning method. Model (A) presents the case when M, the maximum of the arrival sizes is bigger than N, the maximum of the **service** sizes. In Model (B), its dual case N is bigger than M, is treated. In general in Queue models, the state space of the system has the first co-ordinate indicating the number of customers in the system but here the customers in the system are grouped and considered as members of blocks of sizes of the maximum for finding the rate matrix. Using the maximum of the **bulk** arrival size or the maximum of the **bulk** **service** size and grouping the customers as members of blocks in addition to coordinates of the arrival and **service** phases for the partitioning the infinitesimal generator is a new approach in this area. The matrices appearing as the basic system generators in these two models due to block partitioned structure are seen as block circulant matrices. The paper is organized in the following manner. In sections II and III the stationary probability of the number of customers waiting for **service**, the expectation and the variance and the probability of empty queue are derived for these Models (A) and (B). In section IV numerical cases are presented to illustrate them.

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Chaudhry and Templeton [1] have presented extensive discussions of **bulk** **service** systems that operate according to a rule admitting non-accessible batch. Mathias [6] studies a inter-departure time distribution for batches and studies correlation between inter- departure time and batch sizes. Mathias and Alexander [5] provide an discrete-time analysis in the performance evaluation of manufacturing systems. Sivasamy and Elangovan [11] on the other hand, the batch, which entries **service** queue with accessible and non-accessible batches. Goswami, et al., [3] discrete-time **bulk**-**service** queues with accessible and non-accessible batches. Sivasamy and Pukazhenthi [12] have carried out and analyzed the discrete time **bulk** **service** queue for the accessible batch with the arrivals time being geometrical distribution and services time being negative binomial distribution. Vijaya Laxmi Pikala, et.al., [13] have studied the discrete-time renewal input **bulk** **service** queue with changeover time. Pukazhenthi and Ezhilvanan [10] discussed the discrete time queue length distribution with a **bulk** **service** rule. Daniel Wei-Chung Miao1, et al., [2] computational analysis of a markovian queueing system with geometric mean-reverting arrival process. Pukazhenthi and Ezhilvanan [8] the analysis of

sold in various **bulk** sizes depending on market requirements. Noam Paz and Uri Yechali [9] have studied M/M/1 queue with disaster. Usually **bulk** arrival models have M/G/1 upper- Heisenberg block matrix structure. The decomposition of a Toeplitz sub matrix of the infinitesimal generator is required to find the stationary probability vector as done in William J. Stewart [10] and even in such models the recurrence relation method to find the stationary probabilities is stopped at certain level in most general cases indicating limitations of such approach. For M/M/1 **bulk** queues with random environment models one may refer Rama Ganesan, Ramshankar and Ramanarayanan [11]. In this paper the partitioning of the matrix is carried out in a way that the stationary probability vector exhibits a matrix geometric structure for PH/PH/1 **bulk** queues where the arrivals and **service** sizes are finite. Two models (A) and (B) on PH/PH/1 **bulk** queue systems with infinite storage space for customers are studied here using the block partitioning method. In the models considered here, the maximum arrival sizes and the maximum **service** sizes are different. Model (A) presents the case when M, the maximum of arrival sizes is bigger than N, the maximum of the sales sizes. In Model (B), its dual case N is bigger than M, is treated. In general in Queue models, the state space of the system has the first co-ordinate indicating the number of customers in the system but here the customers in the system are grouped and considered as members of M sized blocks of customers when M >N and N sized blocks of customers when N > M for finding the rate matrix. Using the maximum of the **bulk** arrival size or the maximum of the **bulk** **service** size and grouping the customers as members of the blocks for the partitioning the infinitesimal generator is a new approach in this area. The matrices appearing as the basic system generators in these two models due to block partitioned structure are seen as a block circulant matrices. The stationary probability of the number of customers waiting for **service**, the expectation and the variance and the probability of empty queue are derived for these models. Numerical cases are presented to illustrate them.

are very common. Manufactured products arrive in **bulk** sizes and several **bulk** sizes of products are sold in markets. Recently M/M/1 queue system with disaster has been studied by Noam Paz and Uri Yechali [9] but random arrival size or random **service** size with varying environments is not studied. Usually the partitions of the **bulk** arrival models have M/G/1 upper-Heisenberg block matrix structure with zeros below the first sub diagonal. The decomposition of a Toeplitz sub matrix of the infinitesimal generator is required to find the stationary probability vector. Matrix geometric structures have not been noted so far as mentioned by William J. Stewart [10]. But in this paper the partitioning of the matrix is carried out in a way that the stationary probability vectors have a Matrix Geometric solution or a Modified Matrix Geometric solution for infinite capacity C server **bulk** arrival and **bulk** **service** queues with randomly varying environments.

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and **service**-times are assumed to be independent and geometrically distributed. Sivasamy and Pukazhenthi [12] has analyzed the discrete time **bulk** **service** queue with accessible batch with the arrivals and **service** times as general and negative binomially distribution. An interesting paper given by Baburaj [1] and he deals with the concept of a discrete time **bulk** **service** queue under the policy '( , , )′. The inter-arrival times are assumed to be independent and geometrically distributed. Discrete-time renewal input **bulk** **service** queue with changeover time studies has been discussed by Vijaya Laxmi Pikala. et.al., [14]. In this paper they consider a model such that a discrete-time infinite buffer renewal input single server queue with changeover time under ′(a, c, b)′ policy. The **service** and changeover time are geometrically distributed. The server begins **service** if there are at least c customers in the queue and the services are performed in batches of minimum size a and maximum size b (a ≤ c ≤ b). Pukazhenthi and Ezhilvanan [10] discussed the discrete time queue length distribution with a **bulk** **service** rule. Pukazhenthi and Ezhilvanan [11] recently used the analysis of discrete time queues with single server using correlated times.

distribution after the arrival. Model (A) presents the case when M, the maximum of all the maximum arrival sizes in the environments is bigger than N, the maximum of all the maximum **service** sizes in all the environments. In Model (B), its dual, N is bigger than M, is treated. In general in Queue models, the state space of the system has the first co-ordinate indicating the number of customers in the system but here the customers in the system are grouped and considered as members of M sized blocks of customers (when M >N) or N sized blocks of customers (when N > M) for finding the rate matrix. For the C server system under consideration, Model (A) gives three cases namely (A1) M > N ≥ C, (A2) M ≥ C > N and (A3) C > M > N and Model (B) gives two cases namely (B1) N ≥ C, and (B2) C > N. The case M=N with various C values can be treated using Model (A) or Model (B). The matrices appearing as the basic system generators in these models due to block partitions are seen as block circulant matrices. The stationary probability of the number of customers waiting for **service**, the expected queue length, the variance and the probability of empty queue are derived for these models. Numerical cases are presented to illustrate their applications. The paper is organized in the following manner. In section II and section III the BMAP/M/C **bulk** **service** queues with randomly varying environment in which maximum arrival size M is greater than maximum **service** size N and the maximum arrival size M less than the maximum **service** size N are studied respectively with their sub cases. In section IV numerical cases are presented.

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terms of crisp value for FM/M(a,b)/1 under multiple working vacations with fuzzy numbers. The **bulk** **service** rule is applied. The most general **bulk** **service** rule is introduced by Neuts[14]. The batches are served according to FCFS discipline. By considering the arrival rate, **service** rate for busy period, **service** rate for vacation period, exponential distribution of vacation parameters are as trapezoidal fuzzy numbers and the basic idea is to convert all these fuzzy numbers into crisp values by applying Robust Ranking Technique. Further Robust Ranking technique is used to find the expected mean queue length. Moreover, the analytical results are numerically illustrated for L q , P v and P busy under

ABSTRACT : In this paper, a **bulk** arrival general **bulk** **service** queuing system with modified M-vacation policy, variant arrival rate under a restricted admissibility policy of arriving batches and close down time is considered. During the server is in non- vacation, the arrivals are admitted with probability with ' α ' whereas, with probability ' β ' they are admitted when the server is in vacation. The server starts the **service** only if at least ‘a’ customers are waiting in the queue, and renders the **service** according to the general **bulk** **service** rule with minimum of ‘a’ customers and maximum of ‘b’ customers. At the completion of **service**, if the number of waiting customers in the queue is less than ‘𝑎’ then the server performs closedown work , then the server will avail of multiple vacations till the queue length reaches a consecutively avail of M number of vacations, After completing the Mth vacation, if the queue length is still less than a then the server remains idle till it reaches a. The server starts the **service** only if the queue length b ≥ a. It is considered that the variant arrival rate dependent on the state of the server.

A **bulk** arrival general **bulk** **service** queueing with variant threshold policies for secondary jobs is analyzed. The probability generating function for queue size at an arbitrary epoch is derived. Various performance measures are also obtained. Some particular cases are also discussed.

state 0, which means that the arrival distribution is arbitrary when nobody is in the queue. Then A. Y. Chen and E. Renshaw (1997) (2004) established the possibility to let the queue system idle(i.e. nobody is in the queue) at any time. However, all the models discussed before are only the simple queue (i.e. only one person arrives or leaves at the same moment), which eliminates the cases such practical cases as the waiting for the lift and the arrival of the passengers in the aircraft. Since M. F. Neuts (1979) introduced versatile Markovian arrival processes by using several kinds of batch-arrival process and M. L. Chaudhry and J. G. C. Templeton (1983) discussed the first course of **bulk** queues, the theory of batch arrival and **bulk** **service** have been well developed until now. For example, we can see the most recent result from C. Armero and D. Conesa (2000), R. Arumuganathan and K. S. Ramaswami (2005), S. H. Chang, D. W. Choi and T. S. Kim (2004), D. Fakinos (1991), L. Srinivasan, N. Renganathan and R. Kalyanaraman (2002), U. Sumita and Y. Masuda (1997) and P. V. Ushakumari and A. Krishnamoorthy (1998).

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A network consists of a set of interconnected queues. Each queue represents a **service** station, which serves requests (also called jobs) sent by customers. A **service** station consists of one or more servers and a waiting area which holds requests waiting to be served. When a request arrives at a **service** station, its **service** begins immediately if a free server is available. Otherwise, the request is forced to wait in the waiting area. **Service** is done as per the general **bulk** **service** rule introduced by Neuts (1967) with a=1. As per this rule immediately after the completion of the **service**, if the server finds a unit present, it start its **service**; if it finds one or more but almost bit takes them all in a batch and if it finds more than bit takes in the batch for **service** b units , while others wait. The batch takes a minimum of one unit and a maximum of b units.[4]The time between successive request arrivals is called interarrival time. Each request demands a certain amount of **service**,

Specialty clinics provide specialized care for patients referred by primary care physicians, emergency depart- ments, or other specialists. Urgent patients must often be seen on the referral day, while non-urgent referrals are typically booked an appointment for the future. To deliver a balanced performance, the clinics must know how much ‘appointment capacity’ is needed for achieving a reasonably quick access for non-urgent patients. To help identify the capacity that leads to the desired performance, we model the dynamics of appointment backlog as novel discrete-time **bulk** **service** queues, and develop numerical methods for efficient computation of corresponding performance metrics. Realistic features such as arbitrary referral and clinic appointment cancellation distributions, delay-dependent no-show behaviour and rescheduling of no-shows are explicitly captured in our models. The accuracy of the models in predicting performance as well as their usefulness in appointment capacity planning is demonstrated using real data. We also show the application of our models in capacity planning in clinics where patient panel size, rather than appointment capacity, is the major decision variable.

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a mobile proxy node has one **bulk** **service** queue which satisfied the feature of G/G/1 model with a capacity s (all the calculation and analysis in the rest of this paper are based on this assumption). According to queuing theory, in order to deduce the delay and arrival rate, we should figure out the **service** time and waiting time of one piece of message first, which are also relevant with the **service** time and waiting time of **bulk** data.

In this paper production and sales models are studied using matrix geometric methods. Aissani.A and Artalejo.J.R [1] and Ayyappan, Subramanian and Gopal Sekar [2] have analyzed retrial queueing system using matrix geometric methods. Bini, Latouche and Meini [3] have studied numerical methods for Markov chains and Chakravarthy and Neuts [4] have discussed in depth a multi-server queueing model. Gaver, Jacobs and Latouche [5] have treated birth and death models with random environment. Latouche and Ramaswami [6] have studied Analytic methods. For matrix geometric methods and models one may refer Neuts [7]. The models considered here are somewhat different from queueing theory models. Here random number of production and random number of sales of products are considered at a time whereas in queueing theory a fixed number of customers arrive or are served at any arrival or **service** epochs. Even in **bulk** **service** queueing model, when maximum **service** capacity is b, then fixed b customers are cleared by a **service** when more than b customers are waiting, Neuts and Nadarajan [8]. The models considered here being stocks, even when the maximum capacity of the selling system is b, only with some probability, the size b may be cleared. In a similar way arrival sizes of customers are usually fixed in **bulk** arrival queueing systems but the models here treat **bulk** production of products subject to a probability law. Further in these models catastrophic sales can occur, clearing the entire stock of products produced. Such situations are seen

Abstract: The effect of ripening periods on physical, chemical and mechanical properties of **service** tree fruits (Sorbusdomestica L.) were determined. The geometric mean diameter, fruit mass, volume, surface area and fruit density of **service** tree fruits decreased with the increase of ripening periods, while, **bulk** density increased with ripening periods. L*, a*, b*decreased with the increase of ripening periods, while hue increased with ripening periods respectively. The friction coefficients of **service** tree fruits with increase ripening periods were lower for laminate than the other surfaces. The soluble solid content (SSC) and total acidity of **service** tree fruits decreased at ripening period. Titratable acidity was higher in physiological maturity as compared to ripening periods, and pH values of **service** tree fruits decreased with ripening periods. For this reason, post-harvest technological applications of the **service** tree fruits must be designed while taking these criteria into consideration such as physical, mechanical and chemical properties properties of **service** tree fruits.

Various authors have analysed queueing problems of server vacations with several combinationsR. Arumuganathan and T. Judith Malliga analysed of a **bulk** queue with multiple vacations, delayed **service** and setup time[2]. A literature survey on queueing systems with server vacations can be found in Doshi [4]. Reddy et al. [6] have analysed a **bulk** queueing model and multiple vacations with setup time. They derived the expected number of customers in the queue at an arbitrary time epoch and obtained other measures. Recently, Ke [5] has analysed the optimal policy for / /1 queueing systems of different vacation types with startup time.

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[9] S. Maragathasundari and S. Srinivasan, Three phase M/G/1 queue with Bernoulli Feedback and multiple server vacation and **service** interruptions, International Journal of Applied mathematics and Statistics,Vol.33, (2013),Issue no.3.

II. R ADICAL C HANGES A T T HE N ANO S CALE L EVEL When scientists synthesize nanomaterials in the laboratory, everything we understand about a material at a macroscopic level changes at a glance. Color, chemical properties, conductivity - it all changes. At the nanoscale level, the quantum properties of materials overpower its **bulk** properties and stuff gets unusual. The graphite and carbon nanotubes are familiar due to honeycomb (graphene) structure; however, the simple structural change from sheets to tubes changes carbon from one of the softest elements (graphite) to one of the strongest (carbon nanotubes). Gold nanoparticles trade their glossy yellow glitter for a dark reddish shade [3]. The numerous differences between nanoscale and macroscale materials have flung the door wide open for new technology and applications. From medicine to energy to information technology, the nanomaterial revolution is coming.

In Chapter 3, the methods developed to distinguish **bulk** nanobubbles from other nanoparticles are explained in detail. This includes size measurement under the application of external pressure and the determination of nanoparticle density. These methods were applied to armoured nanobubbles that are usually used in ultrasound imaging. The results of these experiments are discussed in detail. These techniques have been applied in chapters 4-6. Chapter 4 investigates the existence of long-lived **bulk** nanobubbles produced using two different devices that are designed to produce **bulk** nanobubble solutions by mechanical means. In Chapter 5, a simple method for generating nanoparticles by mixing ethanol and water is presented. The constitution of these particles whether they are gas filled or otherwise is determined and how the nanoparticles are formed is explained.

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Practical assessment of The U.S. Forest **Service** Forest Health Monitoring (FHM) soil sampling protocol to accurately measure carbon and nitrogen stocks in thousands of forest plots across North America. Twelve established research plots in the French Creek Watershed, of Berks and Chester Counties in Pennsylvania, were revisited in order to obtain sufficient data to compare FHM soil cores to quantitative soil pits on the basis of estimation of **bulk** density, coarse

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