A Duality Relation Between the Workload and Attained Waiting Time in FCFS G/G/s Queues

2013 ◽  
Vol 50 (01) ◽  
pp. 300-307
Author(s):  
Yi-Ching Yao
Keyword(s):  
Stationary Distribution ◽  
Waiting Time ◽  
Busy Period ◽  
Empirical Distribution ◽  
Sample Path ◽  
Duality Relation ◽  
Time Process ◽  
Original Sample ◽  
Reverse Time ◽  
Waiting Time Process

Sengupta (1989) showed that, for the first-come–first-served (FCFS) G/G/1 queue, the workload and attained waiting time of a customer in service have the same stationary distribution. Sakasegawa and Wolff (1990) derived a sample path version of this result, showing that the empirical distribution of the workload values over a busy period of a given sample path is identical to that of the attained waiting time values over the same period. For a given sample path of an FCFS G/G/s queue, we construct a dual sample path of a dual queue which is FCFS G/G/s in reverse time. It is shown that the workload process on the original sample path is identical to the total attained waiting time process on the dual sample path. As an application of this duality relation, we show that, for a time-stationary FCFS M/M/s/k queue, the workload process is equal in distribution to the time-reversed total attained waiting time process.

scholarly journals A Duality Relation Between the Workload and Attained Waiting Time in FCFS G/G/s Queues

2013 ◽  
Vol 50 (1) ◽  
pp. 300-307
Author(s):  
Yi-Ching Yao
Keyword(s):  
Stationary Distribution ◽  
Waiting Time ◽  
Busy Period ◽  
Empirical Distribution ◽  
Sample Path ◽  
Duality Relation ◽  
Time Process ◽  
Original Sample ◽  
Reverse Time ◽  
Waiting Time Process

Sengupta (1989) showed that, for the first-come–first-served (FCFS) G/G/1 queue, the workload and attained waiting time of a customer in service have the same stationary distribution. Sakasegawa and Wolff (1990) derived a sample path version of this result, showing that the empirical distribution of the workload values over a busy period of a given sample path is identical to that of the attained waiting time values over the same period. For a given sample path of an FCFS G/G/s queue, we construct a dual sample path of a dual queue which is FCFS G/G/s in reverse time. It is shown that the workload process on the original sample path is identical to the total attained waiting time process on the dual sample path. As an application of this duality relation, we show that, for a time-stationary FCFS M/M/s/k queue, the workload process is equal in distribution to the time-reversed total attained waiting time process.

A Correlated Queue

1969 ◽  
Vol 6 (01) ◽  
pp. 122-136 ◽  
Author(s):  
B.W. Conolly ◽  
N. Hadidi
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Busy Period ◽  
The State ◽  
Queueing Model ◽  
Time Process ◽  
Waiting Time Process ◽  
The Subject ◽  
Service Pattern ◽  
Arrival Pattern

A “correlated queue” is defined to be a queueing model in which the arrival pattern influences the service pattern or vice versa. A particular model of this nature is considered in this paper. It is such that the service time of a customer is directly proportional to the interval between his own arrival and that of his predecessor. The initial busy period, state and output processes are analyzed in detail. For completeness, a sketch is also given of the analysis of the waiting time process which forms the subject of another paper. The results are used in the analysis of the state and output processes.

The stable M/G/1 queue in heavy traffic and its covariance function

1977 ◽  
Vol 9 (01) ◽  
pp. 169-186 ◽  
Author(s):  
Teunis J. Ott
Keyword(s):  
Brownian Motion ◽  
Waiting Time ◽  
Covariance Function ◽  
Busy Period ◽  
Stationary Process ◽  
Heavy Traffic ◽  
Time Process ◽  
Period Distribution ◽  
Virtual Waiting Time ◽  
Waiting Time Process

Let X(t) be the virtual waiting-time process of a stable M/G/1 queue. Let R(t) be the covariance function of the stationary process X(t), B(t) the busy-period distribution of X(t); and let E(t) = P{X(t) = 0|X(0) = 0}. For X(t) some heavy-traffic results are given, among which are limiting expressions for R(t) and its derivatives and for B(t) and E(t). These results are used to find the covariance function of stationary Brownian motion on [0, ∞).

A Correlated Queue

1969 ◽  
Vol 6 (1) ◽  
pp. 122-136 ◽  
Author(s):  
B.W. Conolly ◽  
N. Hadidi
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Busy Period ◽  
The State ◽  
Queueing Model ◽  
Time Process ◽  
Waiting Time Process ◽  
The Subject ◽  
Service Pattern ◽  
Arrival Pattern

A “correlated queue” is defined to be a queueing model in which the arrival pattern influences the service pattern or vice versa. A particular model of this nature is considered in this paper. It is such that the service time of a customer is directly proportional to the interval between his own arrival and that of his predecessor. The initial busy period, state and output processes are analyzed in detail. For completeness, a sketch is also given of the analysis of the waiting time process which forms the subject of another paper. The results are used in the analysis of the state and output processes.

Limiting diffusions for the conditioned M/G/1 queue

1974 ◽  
Vol 11 (02) ◽  
pp. 355-362 ◽  
Author(s):  
Douglas P. Kennedy
Keyword(s):  
Waiting Time ◽  
Busy Period ◽  
Traffic Intensity ◽  
Time Process ◽  
Brownian Excursion ◽  
Virtual Waiting Time ◽  
Waiting Time Process

The virtual waiting time process, W(t), in the M/G/1 queue is investigated under the condition that the initial busy period terminates but has not done so by time n ≥ t. It is demonstrated that, as n → ∞, W(t), suitably scaled and normed, converges to the unsigned Brownian excursion process or a modification of that process depending whether ρ ≠ 1 or ρ = 1, where ρ is the traffic intensity.

Limiting diffusions for the conditioned M/G/1 queue

1974 ◽  
Vol 11 (2) ◽  
pp. 355-362 ◽  
Author(s):  
Douglas P. Kennedy
Keyword(s):  
Waiting Time ◽  
Busy Period ◽  
Traffic Intensity ◽  
Time Process ◽  
Brownian Excursion ◽  
Virtual Waiting Time ◽  
Waiting Time Process

The virtual waiting time process, W(t), in the M/G/1 queue is investigated under the condition that the initial busy period terminates but has not done so by time n ≥ t. It is demonstrated that, as n → ∞, W(t), suitably scaled and normed, converges to the unsigned Brownian excursion process or a modification of that process depending whether ρ ≠ 1 or ρ = 1, where ρ is the traffic intensity.

The stable M/G/1 queue in heavy traffic and its covariance function

1977 ◽  
Vol 9 (1) ◽  
pp. 169-186 ◽  
Author(s):  
Teunis J. Ott
Keyword(s):  
Brownian Motion ◽  
Waiting Time ◽  
Covariance Function ◽  
Busy Period ◽  
Stationary Process ◽  
Heavy Traffic ◽  
Time Process ◽  
Period Distribution ◽  
Virtual Waiting Time ◽  
Waiting Time Process

Let X(t) be the virtual waiting-time process of a stable M/G/1 queue. Let R(t) be the covariance function of the stationary process X(t), B(t) the busy-period distribution of X(t); and let E(t) = P{X(t) = 0|X(0) = 0}.For X(t) some heavy-traffic results are given, among which are limiting expressions for R(t) and its derivatives and for B(t) and E(t).These results are used to find the covariance function of stationary Brownian motion on [0, ∞).

scholarly journals Periodic queues in heavy traffic

1989 ◽  
Vol 21 (02) ◽  
pp. 485-487 ◽  
Author(s):  
G. I. Falin
Keyword(s):  
Wiener Process ◽  
Waiting Time ◽  
Diffusion Approximation ◽  
Queueing System ◽  
Heavy Traffic ◽  
Time Process ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Periodic Queues ◽  
Poisson Arrivals

An analytic approach to the diffusion approximation in queueing due to Burman (1979) is applied to the M(t)/G/1/∞ queueing system with periodic Poisson arrivals. We show that under heavy traffic the virtual waiting time process can be approximated by a certain Wiener process with reflecting barrier at 0.

The general bulk queue as a matrix factorisation problem of the Wiener-Hopf type. Part II.

1975 ◽  
Vol 7 (3) ◽  
pp. 647-655 ◽  
Author(s):  
John Dagsvik
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Time Process ◽  
Waiting Time Process ◽  
Bulk Queue ◽  
Erlang Distributions ◽  
Matrix Factorisation

In a previous paper (Dagsvik (1975)) the waiting time process of the single server bulk queue is considered and a corresponding waiting time equation is established. In this paper the waiting time equation is solved when the inter-arrival or service time distribution is a linear combination of Erlang distributions. The analysis is essentially based on algebraic arguments.

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