Blocking in a Shared Resource Environment
TL;DR: It is shown that, for the important and commonly implemented policy of complete sharing, a simple one-dimensional recursion can be developed which eliminates all difficulty in computing quantities of interest-regardless of both the size and dimensionality of the underlying model.
Abstract: In recent years, considerable effort has focused on evaluating the blocking experienced by "customers" in contending for a commonly shared "resource." The customers and resource in question have typically been messages and storage space in message storage applications or data streams and bandwidth in data multiplexing applications. The model employed in these studies, a multidimensional generalization of the classical Erlang loss model, has been limited to exponentially distributed storage (or data transmission) times, questions concerning efficient computational schemes have largely been ignored, and the class of resource sharing policies considered has been unnecessarily restricted. The contribution of this paper is threefold. We first show that the state distribution (obtained by previous authors) is valid for the large class of residency time distributions which have rational Laplace transforms. Second, we show that, for the important and commonly implemented policy of complete sharing, a simple one-dimensional recursion can be developed which eliminates all difficulty in computing quantities of interest-regardless of both the size and dimensionality of the underlying model. Third, we show that the state distribution holds for completely arbitrary resource sharing policies.
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TL;DR: The author uses the congestion measures for a multilayer bandwidth-allocation algorithm, emulating some function of virtual circuit setup, fast circuit switching, and fast packet switching at these levels and sheds insight on traffic engineering issues such as appropriate link load, traffic integration, trunk group and switch sizing, and bandwidth reservation criteria for two bursty services.
Abstract: The major benefit of a broadband integrated ATM (asynchronous transfer mode) network is flexible and efficient allocation of communications bandwidth for communications services. However, methods are needed for evaluating congestion for integrated traffic. The author suggests evaluating congestion at different levels, namely the packet level, the burst level, and the call level. Congestion is measured by the probabilities of packet blocking, burst blocking, and call blocking. He outlines the methodologies for comparing these blocking probabilities. The author uses the congestion measures for a multilayer bandwidth-allocation algorithm, emulating some function of virtual circuit setup, fast circuit switching, and fast packet switching at these levels. The analysis also sheds insight on traffic engineering issues such as appropriate link load, traffic integration, trunk group and switch sizing, and bandwidth reservation criteria for two bursty services. >
656 citations
Cites background from "Blocking in a Shared Resource Envir..."
...Kaufman obtained (5.2) for the lossy system W' [ 8 ]....
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TL;DR: It is shown analytically and computationally, that the performance of an optimal pricing strategy is closely matched by a suitably chosen static price, which does not depend on instantaneous congestion, which indicates that the easily implementable time-of-day pricing will often suffice.
Abstract: We consider a service provider (SP) who provides access to a communication network or some other form of on-line services. Users initiate calls that belong to a set of diverse service classes, differing in resource requirements, demand pattern, and call duration. The SP charges a fee per call, which can depend on the current congestion level, and which affects users' demand for calls. We provide a dynamic programming formulation of the problems of revenue and welfare maximization, and derive some qualitative properties of the optimal solution. We also provide a number of approximate approaches, together with an analysis that indicates that near-optimality is obtained for the case of many, relatively small, users. In particular, we show analytically as well as computationally, that the performance of an optimal pricing strategy is closely matched by a suitably chosen static price, which does not depend on instantaneous congestion. This indicates that the easily implementable time-of-day pricing will often suffice. Throughout, we compare the alternative formulations involving revenue or welfare maximization, respectively, and draw some qualitative conclusions.
379 citations
Cites methods from "Blocking in a Shared Resource Envir..."
...[25] J. S. Kaufman, Blocking in a shared resource environment, IEEE Trans....
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..., Kaufman [25] for a method with complexity, or Mitra et al....
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...The loss probabilities satisfies can be efficiently computed; see, e.g., Kaufman [25] for a method with complexity, or Mitra et al. [26] for fast approximations....
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TL;DR: In this article, a mathematical model of a blocking system with simultaneous resource possession is presented, where each customer requests service from one server in each facility in a subset of the service facilities, with the subset depending on the customer class.
Abstract: This paper analyzes a mathematical model of a blocking system with simultaneous resource possession. There are several multiserver service facilities without extra waiting space at which several classes of customers arrive in independent Poisson processes. Each customer requests service from one server in each facility in a subset of the service facilities, with the subset depending on the customer class. If service can be provided immediately upon arrival at all required facilities, then service begins and all servers assigned to the customer start and finish together. Otherwise, the attempt is blocked (lost without generating retrials). The problem is to determine the blocking probability for each customer class. An exact expression is available, but it is complicated. Hence, this paper investigates approximation schemes.
289 citations
TL;DR: The traffic analysis of small-cell mobile networks with dynamic channel assignment is investigated to determine their blocking performance, using a hybrid method of analysis and simulation and significant improvement in network performance is established by numerical results.
Abstract: The traffic analysis of small-cell mobile networks with dynamic channel assignment is investigated to determine their blocking performance, using a hybrid method of analysis and simulation. The authors particularly focus on the performance problems presented by networks with heterogeneous cell traffic loads, the impact of traffic volatility among the cells, and the impact of multichannel traffic on the channel blocking probabilities. Significant improvement in network performance with dynamic channel assignment is established by numerical results. >
277 citations
01 Jan 1985
TL;DR: A mathematical model of a blocking system with simultaneous resource possession, where several classes of customers arrive in independent Poisson processes, is analyzed and approximation schemes are investigated.
Abstract: This paper analyzes a mathematical model of a blocking system with simultaneous resource possession. There are several multiserver service facilities without extra waiting space at which several classes of customers arrive in independent Poisson processes. Each customer requests service from one server in each facility in a subset of the service facilities, with the subset depending on the customer class. If service can be provided immediately upon arrival at all required facilities, then service begins and all servers assigned to the customer start and finish together. Otherwise, the attempt is blocked (lost without generating retrials). The problem is to determine the blocking probability for each customer class. An exact expression is available, but it is complicated. Hence, this paper investigates approximation schemes.
274 citations
Cites background or methods from "Blocking in a Shared Resource Envir..."
...Proof: First, the operator T defined by the right side of (7) obviously maps [0, 1]" into itself....
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...All solutions b* can be bounded above and below, and sometimes found, by successive ap proximation, that is, by iteratively applying the operator T s T\[b(l), ·■·, b(n)]\ mapping [0, 1]" into itself defined by the right side of (7), starting with 1 = (1,1, · · · , 1)....
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...Hence, the successive ap proximation scheme (8) converges in the sense that T(2)*(l) —* L and T(2)*(l) -► U, where L = (Lu ■ ■ ■ , Ln) and U = (Ult ■ ■ ■ , U„) are lower and upper bonds, respectively, on any solution to (7), that is, L(i) s b*(i) < U(i), 1 < i < n....
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...The following theorem indicates that (7) always has a...
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...From (1), we see that (7) yields n polynomial equations in the n unknowns 6*(1), ■ ■ · , 6*(n)....
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References
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TL;DR: Many of the network results of Jackson on arrival and service rate dependencies, of Posner and Bernholtz on different classes of customers, and of Chandy on different types of service centers are combined and extended in this paper.
Abstract: We derive the joint equilibrium distribution of queue sizes in a network of queues containing N service centers and R classes of customers. The equilibrium state probabilities have the general form: P(S) - Cd(S) $f_1$($x_1$)$f_2$($x_2$)...$f_N$($x_N$) where S is the state of the system, $x_i$ is the configuration of customers at the ith service center, d(S) is a function of the state of the model, $f_i$ is a function that depends on the type of the ith service center, and C is a normalizing constant. We consider four types of service centers to model central processors, data channels, terminals, and routing delays. The queueing disciplines associated with these service centers include first-come-first-served, processor sharing, no queueing, and last-come-first-served. Each customer belongs to a single class of customers while awaiting or receiving service at a service center but may change classes and service centers according to fixed probabilities at the completion of a service request. For open networks we consider state dependent arrival processes. Closed networks are those with no arrivals. A network may be closed with respect to some classes of customers and open with respect to other classes of customers. At three of the four types of service centers, the service times of customers are governed by probability distributions having rational Laplace transforms, different classes of customers having different distributions. At first-come-first-served type service centers the service time distribution must be identical and exponential for all classes of customers. Many of the network results of Jackson on arrival and service rate dependencies, of Posner and Bernholtz on different classes of customers, and of Chandy on different types of service centers are combined and extended in this paper. The results become special cases of the model presented here. An example shows how different classes of customers can affect models of computer systems. Finally, we show that an equivalent model encompassing all of the results involves only classes of customers with identical exponentially distributed service times. All of the other structure of the first model can be absorbed into the fixed probabilities governing the change of class and change of service center of each class of customers.
2,416 citations
TL;DR: Methods are presented for computing the equilibrium distribution of customers in closed queueing networks with exponential servers based on two-dimensional iterative techniques which are highly efficient and quite simple to implement.
Abstract: Methods are presented for computing the equilibrium distribution of customers in closed queueing networks with exponential servers. Expressions for various marginal distributions are also derived. The computational algorithms are based on two-dimensional iterative techniques which are highly efficient and quite simple to implement. Implementation considerations such as storage allocation strategies and order of evaluation are examined in some detail.
854 citations
TL;DR: In this article, it was shown that the condition that the service requirements were finite mixtures of gamma distributions was in fact unnecessary, as is shown to be the case in this paper.
Abstract: In a recent article, Kelly [4] has been able to exhibit interesting equilibrium properties for a wide class of ‘quasi-reversible’ queue networks. The assumption of quasi-reversibility puts restrictions on queue discipline, but not on the distributions of the service requirements of customers: however, because of the method of proof he employed, Kelly was forced to impose the condition that the service requirements were finite mixtures of gamma distributions. The form of the results he obtained led him to conjecture that this condition was in fact unnecessary, as is shown to be the case in this paper. The method used to prove the conjecture is of potentially wide application, in problems where the ‘method of stages' leads to useful simplification.
138 citations
TL;DR: The class of queuing networks with multiple routing subchains is extended to include mechanisms of state-dependent lost arrivals and triggered arrivals, and potential applications to modeling computer communication systems with storage and flow control constraints are indicated.
Abstract: The class of queuing networks with multiple routing subchains is extended to include mechanisms of state-dependent lost arrivals and triggered arrivals. A sufficient condition is found, involving the loss and trigger functions, for the equilibrium network state probability distribution to have the product form; the known class of queuing networks with a product form solution is thus enlarged. Such queuing networks are useful models for systems with various population size constraints. Potential applications to modeling computer communication systems with storage and flow control constraints are indicated.
133 citations
TL;DR: Constraints on capacity allocation are investigated for circuit-switched demand access to a common transmission resource by user communities with differing traffic intensity and capacity requirements.
Abstract: Constraints on capacity allocation are investigated for circuit-switched demand access to a common transmission resource by user communities with differing traffic intensity and capacity requirements. A simple birth-death steady-state traffic model is used together with a geometrical representation of any given set of constraints placed upon user access. State probabilities have a simple product form which holds for a wide class of constraint sets. Performance characteristics are intrinsic to the upper surface of the constraint set.
114 citations