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Motivated by growing applications in two-sided markets, we study a parallel matching queue with reneging. Demand and supply units arrive to the system and are matched in an FCFS manner according to a compatibility graph specified by an N-system. If they cannot be matched upon arrival, they queue and may abandon the system as time goes by. We derive explicit product forms of the steady state distributions of this system by identifying a partial balance condition.

We consider a correlated queueing-inventory system with Markovian arrival of customers, phase type distributed service time and Markovian lead time. Items in each cycle have a common life time. Before the realization of this, a purchased item in a cycle can be cancelled in that cycle itself provided inventory level has not dropped to zero. Common life time and inter-cancellation time follow independent exponential distributions. We exhaustively analyze this system. The special case of customer arrival following a Poisson process and service time exponentially distributed, is shown to yield product form solution, thus extending earlier work to the case of correlated lead time. The inventory replenishment policy is to bring the inventory level to its maximum at the lead time realization. Several numerical illustrations are provided to illustrate the system performance.

The purpose of this paper is to obtain product form solution for retrial – queueing – inventory system. We study anM/M/1 retrial queue with a storage system driven by an (s,S) policy. When server is idle, external arrivals enter directly to an orbit. Inventory replenishment lead time is exponentially distributed. The interval between two successive retrials is exponentially distributed and only the customer at the head of the orbit is permitted to access the server. No customer is allowed to join the orbit when the storage system is empty and also when the serer is busy. We first derive the stationary joint distribution of the queue length and the on-hand inventory in explicit product form. Using the joint distribution, we investigate long-run performance measures such as distribution of number of customers served, number of arrivals, number of customers lost during an interval of random duration and a cost function. The optimal pair (s,S) is numerically investigated.

Purpose The use of queueing network models was stimulated by the appearance (1975) of the exact product form solution of a class of open, closed and mixed queueing networks obeying the local balance principle and solved, a few years later, by the popular mean value analysis algorithm (1980). Since then, research efforts have been produced to approximate solutions for non-exponential services and non-pure random mechanisms in customer processing and routing. The purpose of this paper is to examine the suitability of modeling choices and solution approaches consolidated in other domains with respect to two key logistic processes in container terminals. Design/methodology/approach In particular, the analytical solution of queueing networks is assessed for the vessel arrival-departure process and the container internal transfer process with respect to a real terminal of pure transshipment. Findings Numerical experiments show the extent to which a decomposition-based approximation, under fixed or state-dependent arrival rates, may be suitable for the approximate analysis of the queueing network models. Research limitations/implications The limitation of adopting exponential service time distributions and Poisson flows is highlighted. Practical implications Comparisons with a simulation-based solution deliver numerical evidence on the companion use of simulation in the daily practice of managing operations in a finite-time horizon under complex policies. Originality/value Discussion of some open modeling issues and encouraging results provide some guidelines on future research efforts and/or suitable adaption to container terminal logistics of the large body of techniques and algorithms available nowadays for supporting long-run decisions.

We analyse the evolution of a system designed for reservation of some items in advance (for example, seats in aircrafts or trains or bus) by customers arriving at random moments. The reservation has to be done by the server in one of the K time frames. At the beginning of the pth time frame, the inventoried items in it (as well as those sold from it earlier) have life time distribution which is a p-fold convolution of a phase-type distribution with itself, for $$1 \le p \le K.$$ Cancellation of reserved items is possible before the expiry of their life. Distributions characterizing the service and inter-cancellation times are assumed to be independent exponential random variables and the customer arrivals are according to a Markovian arrival process. The number of items for reservation, available at the beginning of each time frame, is finite. If, at the commencement of service of a customer, the item in the required time frame is not available, the reservation may still be possible, through overbooking. Overbooking up to a maximum fixed level is permitted for each time frame. If, for the required day, the overbooked item is available, the customer is served the same. If this too is not available, he is asked to give alternatives. If none of his alternatives can be met, he is provided with a reservation for the time frame (day) for which one is available. If that too is not available, then he will have to wait until the expiry of one time frame; in the last case all remaining customers will have to wait. On expiry of one phase distribution, the time frames are renumbered and a new time frame with a K-fold convolution of the phase-type distribution is added $$(0 \leftarrow 1 \leftarrow 2 \leftarrow \cdots \leftarrow K-1 \leftarrow K \leftarrow K+1).$$ All overbooked customers present in the recently expired time frame are provided with a reservation in the newly added time frame (which has, at that epoch, a life time of a K-fold convolution of the phase-type distribution). This system is analysed and illustrated through numerical experiments. The special case of Poisson arrival, coupled with blocking of arrivals when all time frames $$1, 2, \ldots , K$$ are overbooked, is shown to yield a product form solution. For this case, an appropriate cost function is constructed and its properties investigated numerically.

We consider a queueing-inventory problem with Markovian arrival process, phase type distributed service time. The life time of items is exponentially distributed; all items perish together which means they have a common life time. Reservation (purchase) and cancellation of purchased items is permitted as long as the common life time is not realized. In addition overbooking of items upto a certain maximum is also permitted. When system is fully overbooked waiting customers tend to leave the system. On realization of common life time, instantly S items are procured resulting in the commencement of next cycle. In the particular case of Poisson process and exponentially distributed service time we show that the system admits asymptotic product form solution. System state characteristics are computed which are then numerically illustrated for different sets of values for the input parameters.

International audience; We consider an Energy Packet Network (EPN) model where the service times of the packets follow a Coxian distribution. EPNs are a recent type of models proposed by Gelenbe and his colleagues to study the interactions between energy consumption and the processing or the transmission of data. We prove that such a network has a product form solution for its steady-state distribution. We also state some sufficient conditions for the existence of the flow equations. Finally we study the performances and the energy consumption and we show how to optimize the assignemenl of Δ solar panels over a sensor network.

We consider a link that services multirate quasirandom traffic. Calls are distinguished to handover and new calls. New calls compete for the available bandwidth under a threshold call admission policy. In that policy, new calls of a service-class are blocked if the in-service handover and new calls of the same service-class including the new call, exceeds a predefined threshold. Handover calls compete for the available bandwidth under the complete sharing policy. The steady state probabilities in the proposed model have a product form solution which leads to a convolution algorithm for the accurate calculation of congestion probabilities and link utilization.

International audience; Energy Packet Network (EPN) consists of a queueing network formed by n blocks, where each of them is formed by one data queue, that handles the workload, and one energy queue, that handles packets of energy.We study an EPN model where the energy packets start the transfer. In this model, energy packets are sent to the data queue of the same block. An energy packet routes one workload packet to the next block if the data queue is not empty, and it is lost otherwise.We assume that the energy queues have a finite buffer size and if an energy packet arrives to the system when the buffer is full, jump-over blocking (JOB) is performed, and therefore with some probability it is sent to the data queue and it is lost otherwise.We first provide a value of this probability such that the steady-state probability distribution of packets in the queues admits a product form solution. Moreover, in the case of a single block, we show that the number of data packets in the system decreases as the JOB probability increases.

In this paper, the system under study is a production inventory system that follows (s, S) replenishment policy and having state dependent production rate. The system considered has infinite capacity where customers arrive according to Poisson process. The service time follows exponential distribution. Further in the system, when the inventory level depletes to s, the production process is switched on and is kept on till the inventory level reaches its maximum capacity S. The production time follows exponential distribution with parameter θi, where i represents number of items in the inventory and 0 ≤ i ≤ S − 1. It is assumed that no new customers join the queue when there is void inventory. This yields an explicit product form solution for the steady state probability vector of the system, though there exists a dependence relationship between number of customers joining the queue and time interval for which the production process is turned on. Long run performance measures are computed and lost sales of the system is analysed. A comparison chart that points out the reduction of lost sales with state dependent production rate is also provided along with numerical illustrations for the performance measures. An expected cost function is constructed to numerically investigate the optimal (s, S) pair.

We consider a single-item, two-echelon spare parts inventory model for repairable parts for capital goods with high down time costs. The inventory system consists of a central warehouse and multiple local warehouses, from where customers are served, and a central repair facility at an external supplier. When a part fails at a customer, his request for a ready-for-use part is immediately fullled by his local warehouse if it has a part on stock. At the same time, the failed part is sent to the central repair facility for repair. If the local warehouse is out of stock, then, via an emergency shipment, a ready-for-use part is sent from the central warehouse if it has a part on stock. Otherwise, it is sent via a lateral transshipment from another local warehouse or the external supplier. We assume Poisson demand processes, generally distributed leadtimes for replenishments, repairs, and emergency shipments, and a base-stock policy for the inventory control. Because our inventory system is too complex to solve for a steady-state distribution in closed form, we approximate it by a network of Erlang loss queues with so-called hierarchical jump-over blocking. We show that this network has a steady-state distribution in product-form. Further, this steady-state distribution and several relevant performance measures only depend on the distributions for the repair and replenishment lead times via their means (i.e., they are insensitive for the underlying probability distributions). The steady-state distribution in product-form enables an ecient heuristic for the optimization of base-stock levels, resulting in good approximations of the optimal costs.

The authors propose two multirate teletraffic loss models in a two‐link system that accommodates Poisson arriving calls from different service‐classes with different bandwidth‐per‐call requirements. Each link has two thresholds which refer to the number of in‐service calls in the link. The lowest threshold named support threshold, defines up to which point the link can support calls offloaded from the other link. The highest threshold, named offloading threshold, defines the point where the link starts offloading calls to the other link. Two different bandwidth sharing policies are considered: (i) the complete sharing policy, in which a call can be accepted in a link if there exist enough available bandwidth units and (ii) the bandwidth reservation policy, in which an integer number of bandwidth units is reserved to benefit calls of high bandwidth requirements. The two models do not have a product form solution for the steady state probabilities. However, they propose approximate formulas for the calculation of call blocking probabilities. The accuracy of the formulas is verified through simulation and found to be quite satisfactory.

The authors consider a link accommodating batched Poisson arriving calls of different service classes. A batch of generally distributed number of calls arrive to the link at exponentially distributed time‐points. Each call is treated separately from the rest, and its acceptance is decided, according to the available link bandwidth (partial batch blocking discipline). For congestion control, if the number of in‐service calls of a service class exceeds a threshold (dedicated to the service class), a new call of the same service class is accepted with a probability, dependent on the system state (probabilistic threshold policy). The link is analysed as a multirate loss system, via a reversible Markov chain. The latter leads to a product form solution (PFS) for the steady‐state distribution. Based on the PFS, the authors propose models for the accurate determination of time and call congestion probabilities and link utilisation. Comparison against other existing models under the complete sharing or the bandwidth reservation policy reveals the necessity and consistency of the proposed models.

International audience; Energy Packet Network (EPN) consists of a queueing network formed by N blocks, where each of them is formed by one data queue, that handles the workload, and one energy queue, that handles packets of energy. We study an EPN model where the energy packets start the transfer. In this model, energy packets are sent to the data queue of the same block. An energy packet routes one workload packet to the next block if the data queue is not empty, and it is lost otherwise. We assume that the energy queues have a finite buffer size and if an energy packet arrives to the system when the buffer is full, jump-over blocking (JOB) is performed, and therefore with some probability it is sent to the data queue and it is lost otherwise. We first provide a value of the jump-over blocking probability such that the steadystate probability distribution of packets in the queues admits a product form solution. The product form is established for multiserver and multiclass data packet queues under FCFS, preemptive LCFS and PS discipline. Moreover, in the case of a directed tree queueing network, we show that the number of data packets in each subtree decreases as the JOB probability increases for each block.

Queuing network techniques are effective for evaluating the performance of computer systems. We discuss a queuing network technique for computer systems in finite input source. The finite number of terminals exist in the network and a job in the network moves to the server that includes CPU, I/O equipment and memory after think-time at the terminal. When the job arrives at the server, it acquires a part of memory and executes CPU and I/O processing in the server. After the job completes CPU and I/O processing, it releases the memory and goes back to its original terminal. However, when the computer system has the memory resource, the queuing network model has no product form solution and cannot be calculated the exact solutions. We proposed here an approximation queuing network technique for calculating the performance measures of computer systems with finite input source on which multiple types of jobs exist. This technique involves dividing the queuing network into two levels; one is „inner level„ in which a job executes CPU and I/O processing, and the other is „outer level„ that includes terminals and communication lines. By dividing the network into two levels, we can prevent the number of states of the network from increasing and approximately calculate the performance measures of the network. We evaluated the proposed approximation technique by using numerical experiments and clarified the characteristics of the system response time and the mean number of jobs in the inner level.

In this paper, we consider a production-inventory system with a service facility and production interruptions. Customers arrive in the system according to a Poisson process and require a random time of the service from a single service facility. The service time is assumed to be exponentially distributed. The items are produced according to an (s, S) policy. Each customer leaves the system with one item from the inventory at his service completion epoch if the inventory is available. The production is interrupted for a vacation of random time once the inventory level becomes S. The vacations are exponentially distributed. On return from a vacation, if the inventory level depletes to s, then the production is immediately switched on. It then starts production and is kept in the on mode until the inventory level becomes S. The items in stock are perishable and have exponential life times. It is assumed that no customers is allowed to join the queue when the inventory level is zero. We first derive the stability condition of the system. Then, We obtain the product form solution for the stationary joint distribution of the number of customers and the on-hand inventory level. Based on this stationary distribution, we compute explicitly some performance measures and develop a cost function. Finally, some numerical results are presented.

The power needs of digital devices, their installation in locations where it is difficult to connect them to the power grid and the difficulty of frequently replacing batteries, create the need to operate digital systems with harvested energy. In such cases, local storage batteries must overcome the intermittent nature of the energy supply. System performance then depends on the intermittent energy supply, possible energy leakage, and system workload. Queueing networks with product-form solution (PFS) are standard tools for analyzing the performance of interconnected systems, and predicting relevant performance metrics including job queue lengths, throughput, and system turnaround times and delays. However, existing queueing network models assume unlimited energy availability, whereas intermittently harvested energy can affect system performance due to insufficient energy supply. Thus, this paper develops a new PFS for the joint probability distribution of energy availability, and job queue length for an N-node tandem system. Such models can represent production lines in manufacturing systems, supply chains, cascaded repeaters for optical links, or a data link with multiple input data ports that feeds into a switch or server. Our result enables the rigorous computation of the relevant performance metrics of such systems operating with intermittent energy.

Process flexibility has altered operations in manufacturing and service companies significantly. For instance, auto-mobile manufacturers use flexible production systems to meet uncertain demands effectively, and workforce flexible systems with cross-training are presently common in service industries. This paper studies k-chain configuration in both loss systems and queueing systems. We derive performance measures such as percent of customers loss and average customer waiting time. In the symmetric case, we numerically test the effects of k , system size and traffic intensity on flexibility design. The major conclusion is that 2-chain is no longer effective in loss systems although it still performs well in queueing systems.

In this paper we consider a single server queueing-inventory system having capacity to store S items which have a common-life time (CLT), exponentially distributed with parameter $$\gamma$$ . On realization of $${\textit{CLT}}$$ a replenishment order is placed so as to bring the inventory level back to S, the lead time of which follows exponential distribution with parameter $$\beta$$ . Items remaining are discarded on realization of $${\textit{CLT}}$$ . Customers waiting in the system stay back on realization of common life time. Reservation of items and cancellation of sold items before its expiry time is permitted. Cancellation takes place according to an exponentially distributed inter-occurrence time with parameter $$i\theta$$ when there are $$(S-i)$$ items in the inventory. In this paper we assume that the time required to cancel the reservation is negligible. Customers arrive according to a Poisson process of rate $$\lambda$$ and service time follows exponential distribution with parameter $$\mu$$ . The main assumption that no customer joins the system when inventory level is zero leads to a product form solution of the system state distribution. Several system performance measures are obtained.

A communication link that accommodates different service-classes whose calls have different bandwidth requirements and compete for the available link bandwidth under the Multiple Fractional Channel Reservation (MFCR) policy is considered. The MFCR policy allows the reservation of real (not integer) number of channels in order to benefit calls of high channel (bandwidth) requirements. Two call arrival processes are studied: i) the Poisson (random) process and ii) the quasi-random process. In the first case, calls come from an infinite number of sources, while in the second case calls come from a finite number of sources. To determine call blocking probabilities for Poisson arriving calls, approximate but recursive formulas are proposed based on the notion of reverse transition rates. To determine time and call congestion probabilities for quasi-random arriving calls, approximate but recursive formulas are proved based on the fact that the proposed model does not have a product form solution for the steady state probabilities. The accuracy of the new formulas is verified through simulation and found to be highly satisfactory.

Classical Jackson networks are a well-established tool for the analysis of complex systems. In this paper we analyze Jackson networks with the additional features that (i) nodes may have an infinite supply of low priority work and (ii) nodes may be unstable in the sense that the queue length at these nodes grows beyond any bound. We provide the limiting distribution of the queue length distribution at stable nodes, which turns out to be of product form. A key step in establishing this result is the development of a new algorithm based on adjusted traffic equations for detecting unstable nodes. Our results complement the results known in the literature for the subcases of Jackson networks with either infinite supply nodes or unstable nodes by providing an analysis of the significantly more challenging case of networks with both types of nonstandard node present. Building on our product-form results, we provide closed-form solutions for common customer and system oriented performance measures.

As blood donations are provided on a voluntary non-remunerated basis, blood donors should be treated as user-friendly as possible. Delays and waiting times within blood collection sites (donor centers) should thus be kept at acceptable levels. Waiting times are not incorporated directly other than by practical experience. A more rigorous approach is required. An analytic waiting time computation is therefore investigated to compute waiting times as a function of production. An analytic so-called product form solution for joint queue lengths is concluded. This product form leads to: • an exact expression for the marginal waiting time percentiles at each separate phase • a more formal justification for approximate computation of the total mean waiting time for the non- exponential case. A computational algorithm is provided to numerically approximate the total delay time distribution, an algorithm that has not been presented before. The results are tested for and applied to a real life test case of a Dutch representative blood collection site. These results illustrate the practical usefulness for Sanquin, but also the applicability of the models in general.

In this paper we study a single link, modelled as a loss system, that accommodates Poisson traffic originated from different service-classes. Calls of a service-class are distinguished to new and handover calls. New calls compete for the available link bandwidth under a threshold call admission policy. In this policy, new calls of a service-class are not allowed to enter the system if the number of in-service new and handover calls of the same service-class plus the new call, exceeds a threshold (different for each service-class). On the other hand, handover calls compete for the available link bandwidth under the complete sharing policy. The steady state probabilities in this model have a Product Form Solution (PFS). Due to the existence of a PFS, a convolution algorithm is proposed for the accurate calculation of call blocking probabilities and link utilization.

In this paper, we propose a new multirate teletraffic loss model of a single link with certain bandwidth capacity that accommodates Poisson arriving calls, which can tolerate bandwidth compression, under the threshold policy of admission control. When compression occurs, the service time may increase (elastic calls) or not (adaptive calls). The threshold policy can provide different QoS among service-classes by limiting the number of calls of a service-class up to a predefined threshold, which can be different for each service-class. The proposed model does not have a product form solution for the determination of the steady state probabilities. However, we approximate the model by a reversible Markov chain, and provide recursive formulas for the efficient calculation of the call-level performance metrics, such as call blocking probabilities and link utilization. In addition, we provide similar formulas when the threshold policy co-exists with the bandwidth reservation policy. The latter reserves part of the available link bandwidth to benefit calls of high bandwidth requirements. The accuracy of the proposed formulas is verified through simulation and found to be quite satisfactory. We also show the necessity and consistency of the proposed models.

In this paper we analyze a single server supply chain model in which stocks are kept in both the manufacturer warehouse (production centre) and the retail shop (distribution centre). Arrival of customers to the retail shop form a Poisson process and their service time are exponentially distributed. The maximum stock of the distribution centre is limited to s + Q(=S). When the inventory level depletes to s due to services, it demands Q units at a time from the production centre. The lead time follows an exponential distribution. If the production centre has the required stock on-hand, the items are supplied. Supply of items from the production centre to the distribution centre is done only as a packet of Q units at a time. So if a packet of size Q is not available the distribution centre has to wait till Q units accumulates in the production centre. The production inventory system adopts a (r Q, K Q) policy where the processing of inventory requires a positive random amount of time. Production time for unit item is exponentially distributed. Also we assume that no customer joins the queue when the inventory level in the distribution centre is zero. This assumption leads to an explicit product form solution for the steady state probability vector.

The authors propose a new multirate teletraffic loss model of a single link with certain capacity that accommodates different service-classes whose calls come from finite traffic sources. Calls compete for the available link bandwidth under the combination of the threshold (TH) and the bandwidth reservation (BR) policies. The TH policy can provide different quality of service among service-classes by limiting calls of each service-class up to a certain number, which is a predefined TH, which can be different for each service-class. The BR policy reserves part of the available link bandwidth to benefit calls of high bandwidth requirements. They show that the proposed model, without the BR policy, has a product form solution (PFS) and prove recursive formulas for the efficient calculation of the call-level performance metrics, such as time and call congestion probabilities as well as link utilisation. The combination of the TH and BR policies destroys the PFS of the model. However, they show that approximate but recursive formulas still exist for the efficient calculation of the call-level performance metrics. The accuracy of the proposed formulas is verified through simulation and found to be very satisfactory.

Queueing networks are used to model the performance of the Internet, of manufacturing and job-shop systems, supply chains, and other networked systems in transportation or emergency management. Composed of service stations where customers receive service, and then move to another service station till they leave the network, queueing networks are based on probabilistic assumptions concerning service times and customer movement that represent the variability of system workloads. Subject to restrictive assumptions regarding external arrivals, Markovian movement of customers, and service time distributions, such networks can be solved efficiently with “product form solutions” that reduce the need for software simulators requiring lengthy computations. G-networks generalise these models to include the effect of “signals” that re-route customer traffic, or negative customers that reject service requests, and also have a convenient product form solution. This paper extends G-networks by including a new type of signal, that we call an “Adder”, which probabilistically changes the queue length at the service center that it visits, acting as a load regulator. We show that this generalisation of G-networks has a product form solution.

In this paper, a stochastic approach, based on queueing networks, is analyzed in order to model a supply system, whose nodes are working stations. Unfinished goods and control electrical signals arrive, following Poisson processes, at the nodes. When the working processes at nodes end, according to fixed probabilities, goods can leave the network or move to other nodes as either parts to process or control signals. On the other hand, control signals are activated during a random exponentially distributed time and they act on unfinished parts: precisely, with assigned probabilities, control impulses can move goods between nodes, or destroy them. For the just described queueing network, the stationary state probabilities are found in product form. A numerical algorithm allows to study the steady state probabilities, the mean number of unfinished goods and the stability of the whole network.

A single server retrial queueing-inventory is considered in which customers join directly to the orbit according to a Markovian arrival process (MAP). Service time of customers are independent and identical distributed phase-type distributed (PH) random variables. Inter retrial times are exponentially distributed with parameter \(n\eta \) when n customers are in the orbit. Unsuccessful retrial customers tend to leave the system (impatience) with positive probability. In addition we also introduce search of orbital customers for next service with state dependent probability, immediately on current service completion. This system is shown to be always stable. We compute the long run system state probability. Under certain stringent conditions we prove that a particular case has a product form solution. We get explicit solutions to some retrial queueing models.

As a favorite urban public transport mode, the bike sharing system is a large-scale and complicated system, and there exists a key requirement that a user and a bike should be matched sufficiently in time. Such matched behavior makes analysis of the bike sharing systems more difficult and challenging. To design a better bike sharing system, it is a key to analyze and compute the probabilities of the problematic (i.e., full or empty) stations. In fact, such a computation is established for some fairly complex stochastic systems. To do this, this paper considers a more general large-scale bike sharing system from two important views: (a) Bikes move in an irreducible path graph, which is related to geographical structure of the bike sharing system; and (b) Markovian arrival processes (MAPs) are applied to describe the non-Poisson and burst behavior of bike-user (abbreviated as user) arrivals, while the burstiness demonstrates that the user arrivals are time-inhomogeneous and space-heterogeneous in practice. For such a complicated bike sharing system, this paper establishes a multiclass closed queueing network by means of some virtual ideas, for example, bikes are abstracted as virtual customers; stations and roads are regarded as virtual nodes. Thus user arrivals are related to service times at station nodes; and users riding bikes on roads are viewed as service times at road nodes. Further, to deal with this multiclass closed queueing network, we provide a detailed observation practically on physical behavior of the bike sharing system in order to establish the routing matrix, which gives a nonlinear solution to compute the relative arrival rates in terms of the product-form solution to the steady-state probabilities of joint queue lengths at the virtual nodes. Based on this, we can compute the steady-state probability of problematic stations, and also deal with other interesting performance measures of the bike sharing system. We hope that the methodology and results of this paper can be applicable in the study of more general bike sharing systems through multiclass closed queueing networks.

The purpose of this paper is to obtain product form solution for retrial - queueing - inventory system. We study an M/M/1 retrial queue with a storage system driven by an (s, S) policy. When server is idle, external arrivals enter directly to an orbit. Inventory replenishment lead time is exponentially distributed. The interval between two successive repeat attempts is exponentially distributed and only the customer at the head of the orbit is permitted to access the server. No customer is allowed to join the orbit when the storage system is empty and also when the serer is busy. We first derive the stationary joint distribution of the queue length and the on-hand inventory in explicit product form. Using the joint distribution, we investigate long-run performance measures and a cost function. The optimal pair (s, S) is numerically investigated.

Virtualization of resources is a widely accepted technique to optimize resources in recent technologies. Virtualization allows users to execute their services on the same physical machine, keeping these services isolated from each other. This paper proposes the analytical models for performability evaluation of virtualized servers with dynamic resource utilization. The performance and avalability models are considered separately due to the behaviour of the proposed system. The well-known Markov Reward Model (MRM) is used for the solution of the analytical model considered together with an exact spectral expansion and product form solution. The dynamic resource utilization is employed to enhance the QoS of the proposed model which is another major issue in the performance characterization of virtulazilation. In this paper, the performability output parameters, such as mean queue length, mean response time and blocking probability are computed and presented for the proposed model. In addition, the performability results obtained from the analytical models are validated by the simulation (DES) results to show the accuracy and effectiveness of the proposed work. The results indicate the proposed modelling results show good agreement with DES and understand the factors are very important to improve the QoS.

We study an ( s , S ) production inventory system where the processing of inventory requires a positive random amount of time. As a consequence a queue of demands is formed. Demand process is assumed to be Poisson, duration of each service and time required to add an item to the inventory when the production is on , are independent, non-identically distributed exponential random variables. We assume that no customer joins the queue when the inventory level is zero. This assumption leads to an explicit product form solution for the steady state probability vector, using a simple approach. This is despite the fact that there is a strong correlation between the lead-time (the time required to add an item into the inventory) and the number of customers waiting in the system. The technique is: combine the steady state vector of the classical M/M/1 queue and the steady state vector of a production inventory system where the service is instantaneous and no backlogs are allowed. Using a similar technique, the expected length of a production cycle is also obtained explicitly. The optimal values of S and the production switching on level s have been studied for a cost function involving the steady state system performance measures. Since we have obtained explicit expressions for the performance measures, analytic expressions have been derived for calculating the optimal values of S and s . The technique developed here could be applied to a few other problems in inventory. To substantiate this claim we analyze in detail a variant (which is discussed in Schwarz et al. (2006) ) of the above problem. For that model, we assume that in a production run, production occurs only once in a cycle and the amount produced (in bulk) is sufficient to take the inventory level back to S . A brief discussion on the application of our method to inventory system with lead-time for replenishment has also been provided.