Stationary queue-length and waiting-time distributions in single-server feedback queues

For M/GI/1/∞ queues with instantaneous Bernoulli feedback time- and customer-stationary characteristics of the number of customers in the system and of the waiting time are investigated. Customer-stationary characteristics are thereby obtained describing the behaviour of the queueing processes, for example, at arrival epochs, at feedback epochs, and at times at which an arbitrary (arriving or fed-back) customer enters the waiting room. The method used to obtain these characteristics consists of simple relationships between them and the time-stationary distribution of the number of customers in the system at an arbitrary point in time. The latter is obtained from the wellknown Pollaczek–Khinchine formula for M/GI/1/∞ queues without feedback.

Download Full-text

On two stationary distributions for the stable GI/G/1 queue

Journal of Applied Probability ◽

10.1017/s0021900200118303 ◽

1974 ◽

Vol 11 (04) ◽

pp. 849-852 ◽

Cited By ~ 4

Author(s):

Austin J. Lemoine

Keyword(s):

Waiting Time ◽

Queue Length ◽

Distribution Functions ◽

Single Server ◽

Stationary Distributions ◽

Virtual Waiting Time ◽

Server Systems ◽

Stationary Queue Length

This paper provides simple proofs of two standard results for the stable GI/G/1 queue on the structure of the distribution functions of the stationary virtual waiting time and the stationary queue-length Our argument is applicable to more general single server systems than the queue GI/G/1.

Download Full-text

On two stationary distributions for the stable GI/G/1 queue

Journal of Applied Probability ◽

10.2307/3212570 ◽

1974 ◽

Vol 11 (4) ◽

pp. 849-852 ◽

Cited By ~ 13

Author(s):

Austin J. Lemoine

Keyword(s):

Waiting Time ◽

Queue Length ◽

Distribution Functions ◽

Single Server ◽

Stationary Distributions ◽

Virtual Waiting Time ◽

Server Systems ◽

Stationary Queue Length

This paper provides simple proofs of two standard results for the stable GI/G/1 queue on the structure of the distribution functions of the stationary virtual waiting time and the stationary queue-length Our argument is applicable to more general single server systems than the queue GI/G/1.

Download Full-text

Steady-state analysis of a multiserver queue in the Halfin-Whitt regime

Advances in Applied Probability ◽

10.1239/aap/1214950216 ◽

2008 ◽

Vol 40 (2) ◽

pp. 548-577 ◽

Cited By ~ 14

Author(s):

David Gamarnik ◽

Petar Momčilović

Keyword(s):

Queue Length ◽

Heavy Traffic ◽

Length Distribution ◽

Moment Generating Function ◽

Compact Representation ◽

Single Server ◽

Service Time Distribution ◽

Spare Capacity ◽

Queue Length Distribution ◽

Stationary Queue Length

We consider a multiserver queue in the Halfin-Whitt regime: as the number of serversngrows without a bound, the utilization approaches 1 from below at the rateAssuming that the service time distribution is lattice valued with a finite support, we characterize the limiting scaled stationary queue length distribution in terms of the stationary distribution of an explicitly constructed Markov chain. Furthermore, we obtain an explicit expression for the critical exponent for the moment generating function of a limiting stationary queue length. This exponent has a compact representation in terms of three parameters: the amount of spare capacity and the coefficients of variation of interarrival and service times. Interestingly, it matches an analogous exponent corresponding to a single-server queue in the conventional heavy-traffic regime.

Download Full-text

Decomposition of M/M/1 With Unreliable Service and a Working Vacation

International Journal of Statistics and Probability ◽

10.5539/ijsp.v9n1p63 ◽

2020 ◽

Vol 9 (1) ◽

pp. 63

Author(s):

Joshua Patterson ◽

Andrzej Korzeniowski

Keyword(s):

Closed Form ◽

Failure Rate ◽

Waiting Time ◽

Queue Length ◽

Service Failure ◽

Working Vacation ◽

Stationary Queue Length ◽

Stationary Waiting Time ◽

Relationship Of ◽

The Relationship

We use the stationary distribution for the M/M/1 with Unreliable Service and aWorking Vacation (M/M/1/US/WV) given explicitly in (Patterson & Korzeniowski, 2019) to find a decomposition of the stationary queue length N. By applying the distributional form of Little's Law the Laplace-tieltjes Transform of the stationary customer waiting time W is derived. The closed form of the expected value and variance for both N and W is found and the relationship of the expected stationary waiting time as a function of the service failure rate is determined.

Download Full-text

Convexity results for single-server queues and for multiserver queues with constant service times

Journal of Applied Probability ◽

10.1017/s0021900200038948 ◽

1990 ◽

Vol 27 (02) ◽

pp. 465-468 ◽

Cited By ~ 1

Author(s):

Arie Harel

Keyword(s):

Waiting Time ◽

Service Time ◽

Sojourn Time ◽

Interarrival Time ◽

Service Rate ◽

Single Server ◽

Multiserver Queues ◽

Service Rates ◽

Constant Service Times ◽

Number Of Customers

We show that the waiting time in queue and the sojourn time of every customer in the G/G/1 and G/D/c queue are jointly convex in mean interarrival time and mean service time, and also jointly convex in mean interarrival time and service rate. Counterexamples show that this need not be the case, for the GI/GI/c queue or for the D/GI/c queue, for c ≧ 2. Also, we show that the average number of customers in the M/D/c queue is jointly convex in arrival and service rates. These results are surprising in light of the negative result for the GI/GI/2 queue (Weber (1983)).

Download Full-text

The Trivariate Distribution of the Maximum Queue Length, the Number of Customers Served and the Duration of the Busy Period for the M/G/1 Queueing System

Journal of Applied Probability ◽

10.2307/3212282 ◽

1969 ◽

Vol 6 (1) ◽

pp. 154-161 ◽

Cited By ~ 12

Author(s):

E.G. Enns

Keyword(s):

Queue Length ◽

Busy Period ◽

Queueing System ◽

Single Server ◽

List Type ◽

Type Number ◽

Distribution Of The Maximum ◽

Maximum Queue Length ◽

Number Of Customers

In the study of the busy period for a single server queueing system, three variables that have been investigated individually or at most in pairs are:1.The duration of the busy period.2.The number of customers served during the busy period.3.The maximum number of customers in the queue during the busy period.

Download Full-text

A single server tandem queue

Journal of Applied Probability ◽

10.2307/3211840 ◽

1971 ◽

Vol 8 (1) ◽

pp. 95-109 ◽

Cited By ~ 16

Author(s):

Sreekantan S. Nair

Keyword(s):

Waiting Time ◽

Queue Length ◽

Busy Period ◽

Single Server ◽

Queueing Model ◽

Tandem Queue ◽

Limiting Behavior ◽

Virtual Waiting Time ◽

Priority Model

Avi-Itzhak, Maxwell and Miller (1965) studied a queueing model with a single server serving two service units with alternating priority. Their model explored the possibility of having the alternating priority model treated in this paper with a single server serving alternately between two service units in tandem.Here we study the distribution of busy period, virtual waiting time and queue length and their limiting behavior.

Download Full-text

Convexity results for single-server queues and for multiserver queues with constant service times

Journal of Applied Probability ◽

10.2307/3214668 ◽

1990 ◽

Vol 27 (2) ◽

pp. 465-468 ◽

Cited By ~ 5

Author(s):

Arie Harel

Keyword(s):

Waiting Time ◽

Service Time ◽

Sojourn Time ◽

Interarrival Time ◽

Service Rate ◽

Single Server ◽

Multiserver Queues ◽

Service Rates ◽

Constant Service Times ◽

Number Of Customers

We show that the waiting time in queue and the sojourn time of every customer in the G/G/1 and G/D/c queue are jointly convex in mean interarrival time and mean service time, and also jointly convex in mean interarrival time and service rate. Counterexamples show that this need not be the case, for the GI/GI/c queue or for the D/GI/c queue, for c ≧ 2. Also, we show that the average number of customers in the M/D/c queue is jointly convex in arrival and service rates.These results are surprising in light of the negative result for the GI/GI/2 queue (Weber (1983)).

Download Full-text

The probability of large queue lengths and waiting times in a heterogeneous multiserver queue II: Positive recurrence and logarithmic limits

Advances in Applied Probability ◽

10.2307/1427839 ◽

1995 ◽

Vol 27 (2) ◽

pp. 567-583 ◽

Cited By ~ 13

Author(s):

John S. Sadowsky

Keyword(s):

Waiting Time ◽

Queue Length ◽

Waiting Times ◽

General Setting ◽

Batch Arrival ◽

Positive Recurrence ◽

Waiting Time Distributions ◽

Harris Recurrence ◽

Stationary Queue Length ◽

Queue Lengths

We continue our investigation of the batch arrival-heterogeneous multiserver queue begun in Part I. In a general setting we prove the positive Harris recurrence of the system, and with no additional conditions we prove logarithmic tail limits for the stationary queue length and waiting time distributions.

Download Full-text