scholarly journals Loss Systems with Slow Retrials in the Halfin–Whitt Regime

2013 ◽  
Vol 45 (1) ◽  
pp. 274-294 ◽  
Author(s):  
F. Avram ◽  
A. J. E. M. Janssen ◽  
J. S. H. Van Leeuwaarden
Keyword(s):  
Steady State ◽  
Critical Load ◽  
Performance Measures ◽  
System Performance ◽  
Economies Of Scale ◽  
Blocking Probability ◽  
Loss System ◽  
Qed Regime ◽  
Loss Systems ◽  
Steady State Performance

The Halfin–Whitt regime, or the quality-and-efficiency-driven (QED) regime, for multiserver systems refers to a situation with many servers, a critical load, and yet favorable system performance. We apply this regime to the classical multiserver loss system with slow retrials. We derive nondegenerate limiting expressions for the main steady-state performance measures, including the retrial rate and the blocking probability. It is shown that the economies of scale associated with the QED regime persist for systems with retrials, although in situations when the load becomes extremely critical the retrials cause deteriorated performance. Most of our results are obtained by a detailed analysis of Cohen's equation that defines the retrial rate in an implicit way. The limiting expressions are established by studying prelimit behavior and exploiting the connection between Cohen's equation and Mills' ratio for the Gaussian and Poisson distributions.

scholarly journals Loss Systems with Slow Retrials in the Halfin–Whitt Regime

2013 ◽  
Vol 45 (01) ◽  
pp. 274-294 ◽  
Author(s):  
F. Avram ◽  
A. J. E. M. Janssen ◽  
J. S. H. Van Leeuwaarden
Keyword(s):  
Steady State ◽  
Critical Load ◽  
Performance Measures ◽  
System Performance ◽  
Economies Of Scale ◽  
Blocking Probability ◽  
Loss System ◽  
Qed Regime ◽  
Loss Systems ◽  
Steady State Performance

The Halfin–Whitt regime, or the quality-and-efficiency-driven (QED) regime, for multiserver systems refers to a situation with many servers, a critical load, and yet favorable system performance. We apply this regime to the classical multiserver loss system with slow retrials. We derive nondegenerate limiting expressions for the main steady-state performance measures, including the retrial rate and the blocking probability. It is shown that the economies of scale associated with the QED regime persist for systems with retrials, although in situations when the load becomes extremely critical the retrials cause deteriorated performance. Most of our results are obtained by a detailed analysis of Cohen's equation that defines the retrial rate in an implicit way. The limiting expressions are established by studying prelimit behavior and exploiting the connection between Cohen's equation and Mills' ratio for the Gaussian and Poisson distributions.

Sensitivity analysis for stationary and ergodic queues

1992 ◽  
Vol 24 (03) ◽  
pp. 738-750 ◽  
Author(s):  
P. Konstantopoulos ◽  
Michael A. Zazanis
Keyword(s):  
Sensitivity Analysis ◽  
Steady State ◽  
Performance Measures ◽  
Perturbation Analysis ◽  
Service Process ◽  
Single Server ◽  
Stationary Version ◽  
Major Interest ◽  
Regenerative Approach ◽  
Steady State Performance

Starting with some mild assumptions on the parametrization of the service process, perturbation analysis (PA) estimates are obtained for stationary and ergodic single-server queues. Besides relaxing the stochastic assumptions, our approach solves some problems associated with the traditional regenerative approach taken in most of the previous work in this area. First, it avoids problems caused by perturbations interfering with the regenerative structure of the system. Second, given that the major interest is in steady-state performance measures, it examines directly the stationary version of the system, instead of considering performance measures expressed as Cesaro limits. Finally, it provides new estimators for general (possibly discontinuous) functions of the workload and other steady-state quantities.

A note on a D/G/K loss system with retrials

1990 ◽  
Vol 27 (2) ◽  
pp. 385-392 ◽  
Author(s):  
Behnam Pourbabai
Keyword(s):  
Steady State ◽  
Processing Time ◽  
Distributed Processing ◽  
Random Access ◽  
Heterogeneous Servers ◽  
Loss System ◽  
System A ◽  
Loss Systems

An algorithm is suggested for approximating the performance of a D/G/K loss system with deterministic input, generally distributed processing time, K heterogeneous servers, the random access processing discipline, and retrials in steady state. In loss systems with retrials, the units which at the instants of their arrival at the system find all the servers busy, are not lost: those units retry to be processed by merging with the incoming arrival units. In this system, a fraction of the units which have not initially been processed will be allowed to leave the system. The performance of this system in steady state is approximated by a recursive technique.

A monotonicity result for a single-server loss system

1995 ◽  
Vol 32 (04) ◽  
pp. 1112-1117
Author(s):  
Xiuli Chao ◽  
Liyi Dai
Keyword(s):  
Markov Process ◽  
Infinitesimal Generator ◽  
Blocking Probability ◽  
Matrix Analysis ◽  
Service Process ◽  
Single Server ◽  
Loss System ◽  
Special Cases ◽  
Monotonicity Result ◽  
Loss Systems

We consider a family of M(t)/M(t)/1/1 loss systems with arrival and service intensities (λt (c), μt (c)) = (λct , μct ), where (λt , μt ) are governed by an irreducible Markov process with infinitesimal generator Q = (qij )m × m such that (λt , μt ) = (λi , μi ) when the Markov process is in state i. Based on matrix analysis we show that the blocking probability is decreasing in c in the interval [0, c ∗], where c ∗ = 1/maxi Σ j ≠i qij /(λi + μi ). Two special cases are studied for which the result can be extended to all c. These results support Ross's conjecture that a more regular arrival (and service) process leads to a smaller blocking probability.

Perishable Inventory System with Bi-Level Service System

2016 ◽  
pp. 200-217
Author(s):  
D. Gomathi
Keyword(s):  
Steady State ◽  
Performance Measures ◽  
Service Time ◽  
System Performance ◽  
Service System ◽  
Joint Probability ◽  
Inventory System ◽  
Service Time Distribution ◽  
Perishable Inventory ◽  
Perishable Inventory System

In this chapter we consider a perishable inventory system under continuous review at a bi-level service system with finite waiting hall of size N. The maximum storage capacity of the inventory is S units. We assumed that a demand for the commodity is of unit size. The arrival time points of customers form a Poisson process. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. The effect of the two modes of operations on the system performance measures is also discussed. It is also assumed that lead time for the reorders is distributed as exponential and is independent of the service time distribution. The items are perishable in nature and the life time of each item is assumed to be exponentially distributed. The demands that occur during stock out periods are lost. The joint probability distribution of the number of customers is obtained in the steady-state case. Various system performance measures in the steady state are derived. The results are illustrated numerically.

Sensitivity analysis for stationary and ergodic queues

1992 ◽  
Vol 24 (3) ◽  
pp. 738-750 ◽  
Author(s):  
P. Konstantopoulos ◽  
Michael A. Zazanis
Keyword(s):  
Sensitivity Analysis ◽  
Steady State ◽  
Performance Measures ◽  
Perturbation Analysis ◽  
Service Process ◽  
Single Server ◽  
Stationary Version ◽  
Major Interest ◽  
Regenerative Approach ◽  
Steady State Performance

Starting with some mild assumptions on the parametrization of the service process, perturbation analysis (PA) estimates are obtained for stationary and ergodic single-server queues. Besides relaxing the stochastic assumptions, our approach solves some problems associated with the traditional regenerative approach taken in most of the previous work in this area. First, it avoids problems caused by perturbations interfering with the regenerative structure of the system. Second, given that the major interest is in steady-state performance measures, it examines directly the stationary version of the system, instead of considering performance measures expressed as Cesaro limits. Finally, it provides new estimators for general (possibly discontinuous) functions of the workload and other steady-state quantities.

A note on a D/G/K loss system with retrials

1990 ◽  
Vol 27 (02) ◽  
pp. 385-392
Author(s):  
Behnam Pourbabai
Keyword(s):  
Steady State ◽  
Processing Time ◽  
Distributed Processing ◽  
Random Access ◽  
Heterogeneous Servers ◽  
Loss System ◽  
System A ◽  
Loss Systems

An algorithm is suggested for approximating the performance of a D/G/K loss system with deterministic input, generally distributed processing time, K heterogeneous servers, the random access processing discipline, and retrials in steady state. In loss systems with retrials, the units which at the instants of their arrival at the system find all the servers busy, are not lost: those units retry to be processed by merging with the incoming arrival units. In this system, a fraction of the units which have not initially been processed will be allowed to leave the system. The performance of this system in steady state is approximated by a recursive technique.

Periodic steady state of loss systems with periodic inputs

1998 ◽  
Vol 30 (1) ◽  
pp. 152-166 ◽  
Author(s):  
Helmut Willie
Keyword(s):  
Steady State ◽  
Marked Point ◽  
Sufficient Conditions ◽  
Initial Distribution ◽  
Marked Point Process ◽  
Loss System ◽  
State Behaviour ◽  
Loss Systems ◽  
Periodic Steady State ◽  
Periodic Inputs

The input of a multiserver loss system is assumed to be a periodic random marked point process which has, with probability one, infinitely many construction points. It is shown that, independently of the initial distribution, there exists a unique periodic process modeling the periodic steady-state behaviour of the loss system. In addition, practical sufficient conditions for the existence of enough construction points are derived.

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