An M/G/1 model with finite waiting room in which a customer remains during part of service

1973 ◽  
Vol 10 (4) ◽  
pp. 778-785 ◽  
Author(s):  
Stig I. Rosenlund
Keyword(s):  
Service System ◽  
Asymptotic Expression ◽  
Length Distribution ◽  
Loss Ratio ◽  
Efficiency Measure ◽  
Waiting Room ◽  
System Of Integral Equations ◽  
Large Intensity ◽  
Long Run ◽  
Stieltjes Transforms

An M/G/1 service system with finite waiting room is studied. A customer is served by one server in phases, during some of which a place in the waiting room is occupied. The busy period length distribution is obtained from a system of integral equations leading to a linear system in Laplace-Stieltjes transforms. An asymptotic expression, for large intensity of arrival, for the expectation of this length is given. An efficiency measure giving the long run customer loss ratio is obtained. The model is shown to apply to an inventory and a container traffic problem.

An M/G/1 model with finite waiting room in which a customer remains during part of service

1973 ◽  
Vol 10 (04) ◽  
pp. 778-785 ◽  
Author(s):  
Stig I. Rosenlund
Keyword(s):  
Service System ◽  
Asymptotic Expression ◽  
Length Distribution ◽  
Loss Ratio ◽  
Efficiency Measure ◽  
Waiting Room ◽  
System Of Integral Equations ◽  
Large Intensity ◽  
Long Run ◽  
Stieltjes Transforms

An M/G/1 service system with finite waiting room is studied. A customer is served by one server in phases, during some of which a place in the waiting room is occupied. The busy period length distribution is obtained from a system of integral equations leading to a linear system in Laplace-Stieltjes transforms. An asymptotic expression, for large intensity of arrival, for the expectation of this length is given. An efficiency measure giving the long run customer loss ratio is obtained. The model is shown to apply to an inventory and a container traffic problem.

Busy periods in time-dependent M/G/1 queues

1976 ◽  
Vol 8 (1) ◽  
pp. 195-208 ◽  
Author(s):  
Stig. I. Rosenlund
Keyword(s):  
Integral Representations ◽  
Single Server ◽  
Waiting Room ◽  
Batch Arrivals ◽  
System Of Integral Equations ◽  
Long Run ◽  
Asymptotic Expressions ◽  
In Series ◽  
Cramer’S Rule ◽  
Non Homogeneous Poisson Process

A single-server queue with batch arrivals in a non-homogeneous Poisson process and with balking is studied with respect to the busy period, using supplementary variables. A system of integral equations is obtained on the base of which the transforms are expressed in series. For the homogeneous case, assuming finite waiting room, the solutions are obtained via Cramer's rule. This gives asymptotic expressions for the expectations for large arrival intensity. An efficiency measure giving the long run loss probability is given. For a special case contour integral representations are given as solutions.

Busy periods in time-dependent M/G/1 queues

1976 ◽  
Vol 8 (01) ◽  
pp. 195-208 ◽  
Author(s):  
Stig. I. Rosenlund
Keyword(s):  
Integral Representations ◽  
Single Server ◽  
Waiting Room ◽  
Batch Arrivals ◽  
System Of Integral Equations ◽  
Long Run ◽  
Asymptotic Expressions ◽  
In Series ◽  
Cramer’S Rule ◽  
Non Homogeneous Poisson Process

A single-server queue with batch arrivals in a non-homogeneous Poisson process and with balking is studied with respect to the busy period, using supplementary variables. A system of integral equations is obtained on the base of which the transforms are expressed in series. For the homogeneous case, assuming finite waiting room, the solutions are obtained via Cramer's rule. This gives asymptotic expressions for the expectations for large arrival intensity. An efficiency measure giving the long run loss probability is given. For a special case contour integral representations are given as solutions.

Throughput maximization in a loss queueing system with heterogeneous servers

1990 ◽  
Vol 27 (03) ◽  
pp. 693-700 ◽  
Author(s):  
Matthew J. Sobel
Keyword(s):  
Queueing System ◽  
Blocking Probability ◽  
Cost Structure ◽  
Throughput Maximization ◽  
Queueing Model ◽  
Heterogeneous Servers ◽  
Waiting Room ◽  
Discounted Cost ◽  
Long Run ◽  
Poisson Arrivals

Assigning each arriving customer to the fastest idle server is shown to maximize throughput (equivalently, minimize blocking probability) in a queueing model with Poisson arrivals, heterogeneous exponential servers, and no waiting room. If a cost structure is imposed on this model, under specified conditions the same policy minimizes the expected discounted cost and the long-run average cost per unit time.

FORECAST ERRORS IN SERVICE SYSTEMS

2009 ◽  
Vol 23 (2) ◽  
pp. 305-332 ◽  
Author(s):  
Samuel G. Steckley ◽  
Shane G. Henderson ◽  
Vijay Mehrotra
Keyword(s):  
System Analysis ◽  
Service System ◽  
Arrival Rate ◽  
Forecast Errors ◽  
Normal Mixture ◽  
Large Dataset ◽  
Long Run ◽  
Great Generality ◽  
Short Run ◽  
Timely Fashion

We investigate the presence and impact of forecast errors in the arrival rate of customers to a service system. Analysis of a large dataset shows that forecast errors can be large relative to the fluctuations naturally expected in a Poisson process. We show that ignoring forecast errors typically leads to overestimates of performance and that forecast errors of the magnitude seen in our dataset can have a practically significant impact on predictions of long-run performance. We also define short-run performance as the random percentage of calls received in a particular period that are answered in a timely fashion. We prove a central limit theorem that yields a normal-mixture approximation for its distribution for Markovian queues and we sketch an argument that shows that a normal-mixture approximation should be valid in great generality. Our results provide motivation for studying staffing strategies that are more flexible than the fixed-level staffing rules traditionally studied in the literature.

scholarly journals Congestion-dependent pricing in a stochastic service system

2007 ◽  
Vol 39 (04) ◽  
pp. 898-921 ◽  
Author(s):  
Idriss Maoui ◽  
Hayriye Ayhan ◽  
Robert D. Foley
Keyword(s):  
Finite Number ◽  
Service Provider ◽  
Queueing System ◽  
Service System ◽  
Pricing Policy ◽  
Holding Costs ◽  
State Dependent ◽  
Long Run ◽  
Number Of Classes ◽  
Over Time

We study a service facility modeled as a queueing system with finite or infinite capacity. Arriving customers enter if there is room in the facility and if they are willing to pay the price posted by the service provider. Customers belong to one of a finite number of classes that have different willingnesses-to-pay. Moreover, there is a penalty for congestion in the facility in the form of state-dependent holding costs. The service provider may advertise class-specific prices that may fluctuate over time. We show the existence of a unique optimal stationary pricing policy in a continuous and unbounded action space that maximizes the long-run average profit per unit time. We determine an expression for this policy under certain conditions. We also analyze the structure and the properties of this policy.

scholarly journals Asymptotic analysis of a queueing system by a two-dimensional state space

1984 ◽  
Vol 21 (4) ◽  
pp. 870-886 ◽  
Author(s):  
J. P. C. Blanc
Keyword(s):  
Asymptotic Analysis ◽  
Queue Length ◽  
Queueing System ◽  
Queueing Systems ◽  
Length Distribution ◽  
Time Dependent ◽  
Queue Length Distribution ◽  
Long Run ◽  
Dimensional State Space ◽  
Hilbert Type

A technique is developed for the analysis of the asymptotic behaviour in the long run of queueing systems with two waiting lines. The generating function of the time-dependent joint queue-length distribution is obtained with the aid of the theory of boundary value problems of the Riemann–Hilbert type and by introducing a conformal mapping of the unit disk onto a given domain. In the asymptotic analysis an extensive use is made of theorems on the boundary behaviour of such conformal mappings.

scholarly journals Asymptotic analysis of a queueing system by a two-dimensional state space

1984 ◽  
Vol 21 (04) ◽  
pp. 870-886
Author(s):  
J. P. C. Blanc
Keyword(s):  
Asymptotic Analysis ◽  
Queue Length ◽  
Queueing System ◽  
Queueing Systems ◽  
Length Distribution ◽  
Time Dependent ◽  
Queue Length Distribution ◽  
Long Run ◽  
Dimensional State Space ◽  
Hilbert Type

A technique is developed for the analysis of the asymptotic behaviour in the long run of queueing systems with two waiting lines. The generating function of the time-dependent joint queue-length distribution is obtained with the aid of the theory of boundary value problems of the Riemann–Hilbert type and by introducing a conformal mapping of the unit disk onto a given domain. In the asymptotic analysis an extensive use is made of theorems on the boundary behaviour of such conformal mappings.

Queues with path-dependent arrival processes

2021 ◽  
Vol 58 (2) ◽  
pp. 484-504
Author(s):  
Kerry Fendick ◽  
Ward Whitt
Keyword(s):  
Point Processes ◽  
Heavy Traffic ◽  
Length Distribution ◽  
Arrival Process ◽  
Diffusion Limit ◽  
Limiting Behavior ◽  
Queue Length Distribution ◽  
Long Run ◽  
Arrival Processes ◽  
Path Dependent

AbstractWe study the transient and limiting behavior of a queue with a Pólya arrival process. The Pólya process is interesting because it exhibits path-dependent behavior, e.g. it satisfies a non-ergodic law of large numbers: the average number of arrivals over time [0, t] converges almost surely to a nondegenerate limit as $t \rightarrow \infty$. We establish a heavy-traffic diffusion limit for the $\sum_{i=1}^{n} P_i/GI/1$ queue, with arrivals occurring exogenously according to the superposition of n independent and identically distributed Pólya point processes. That limit yields a tractable approximation for the transient queue-length distribution, because the limiting net input process is a Gaussian Markov process with stationary increments. We also provide insight into the long-run performance of queues with path-dependent arrival processes. We show how Little’s law can be stated in this context, and we provide conditions under which there is stability for a queue with a Pólya arrival process.

A service system with two stages of waiting and feedback of customers

1984 ◽  
Vol 21 (02) ◽  
pp. 404-413 ◽  
Author(s):  
Osman M. E. Ali ◽  
Marcel F. Neuts
Keyword(s):  
Service System ◽  
Waiting Times ◽  
Random Numbers ◽  
Stationary Distributions ◽  
Waiting Room ◽  
Service Unit ◽  
Queue Lengths ◽  
Two Stages

Customers initially enter a service unit via a waiting room. The customers to be served are stored in a service room which is replenished by the transfer of all those in the waiting room at the points in time where the service room becomes empty. At those epochs of transfer, positive random numbers of ‘overhead customers' are also added to the service room. Algorithmically tractable expressions for the stationary distributions of queue lengths and waiting times at various embedded random epochs are derived. The discussion generalizes an earlier treatment by Takács in several directions.

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