scholarly journals The theory of networks of single server queues and the tandem queue model

1997 ◽  
Vol 10 (4) ◽  
pp. 363-381
Author(s):  
Pierre Le Gall
Keyword(s):  
Traffic Simulation ◽  
Heavy Traffic ◽  
Telecommunication Networks ◽  
Single Server ◽  
Tandem Queue ◽  
Single Server Queues ◽  
Queueing Delay ◽  
Jitter Delay ◽  
Service Times ◽  
Highly Correlated

We consider the stochastic behavior of networks of single server queues when successive service times of a given customer are highly correlated. The study is conducted in two particular cases: 1) networks in heavy traffic, and 2) networks in which all successive service times have the same value (for a given customer), in order to avoid the possibility of breaking up the busy periods. We then show how the local queueing delay (for an arbitrary customer) can be derived through an equivalent tandem queue on the condition that one other local queueing delay is added: the jitter delay due to the independence of partial traffic streams.We consider a practical application of the results by investigating the influence of long packets on the queueing delay of short packets in modern packet switched telecommunication networks. We compare these results with the results given by traffic simulation methods to conclude that there is good agreement between results of calculation and of traffic simulation.

scholarly journals Single server queueing networks with varying service times and renewal input

2000 ◽  
Vol 13 (4) ◽  
pp. 429-450 ◽  
Author(s):  
Pierre Le Gall
Keyword(s):  
Network Management ◽  
Queueing Networks ◽  
Queueing Network ◽  
Single Server ◽  
Simulation Methods ◽  
Tandem Queues ◽  
Tandem Queue ◽  
Queueing Delay ◽  
New Concepts ◽  
Service Times

Using recent results in tandem queues and queueing networks with renewal input, when successive service times of the same customer are varying (and when the busy periods are frequently not broken up in large networks), the local queueing delay of a single server queueing network is evaluated utilizing new concepts of virtual and actual delays (respectively). It appears that because of an important property, due to the underlying tandem queue effect, the usual queueing standards (related to long queues) cannot protect against significant overloads in the buffers due to some possible “agglutination phenomenon” (related to short queues). Usual network management methods and traffic simulation methods should be revised, and should monitor the partial traffic streams loads (and not only the server load).

scholarly journals Single server tandem queues and queueing networks with non-correlated successive service times

2001 ◽  
Vol 14 (4) ◽  
pp. 381-398
Author(s):  
Pierre Le Gall
Keyword(s):  
Sojourn Time ◽  
Queueing Networks ◽  
Interarrival Time ◽  
Single Server ◽  
Tandem Queue ◽  
Traditional Concept ◽  
Queueing Delay ◽  
Service Times ◽  
Two Stages ◽  
Traffic Source

To evaluate the local actual queueing delay in general single server queueing networks with non-correlated successive service times for the same customer, we start from a recent work using the tandem queue effect, when two successive local arrivals are not separated by “premature departures”. In that case, two assumptions were made: busy periods not broken up, and there are limited variations for successive service times. These assumptions are given up after having crossed two stages. The local arrivals become indistinguishable for the sojourn time inside a given busy period. It is then proved that the local sojourn time of this tandem queue effect may be considered as the sum of two components: the first (independent of the local interarrival time) corresponding to the case where upstream, successive service times are supposed to be identical to the local service time, and the second (negligible after having crossed 2 or 3 stages) depending on local interarrival times increasing because of broken up busy periods. The consequence is the possible occurrence of the agglutination phenomenon of indistinguishable customers in the buffers (when there are limited “premature departures”), due to a stronger impact of long service times upon the local actual queueing delay, which is not consistent with the traditional concept of local traffic source only generating distinguishable customers.

On a tandem queueing model with identical service times at both counters, II

1979 ◽  
Vol 11 (3) ◽  
pp. 644-659 ◽  
Author(s):  
O. J. Boxma
Keyword(s):  
Heavy Traffic ◽  
Queueing Systems ◽  
Single Server ◽  
Service Time Distribution ◽  
In Series ◽  
Mean And Variance ◽  
Queues In Series ◽  
The Mean ◽  
Service Times ◽  
Practical Implications

This paper is devoted to the practical implications of the theoretical results obtained in Part I [1] for queueing systems consisting of two single-server queues in series in which the service times of an arbitrary customer at both queues are identical. For this purpose some tables and graphs are included. A comparison is made—mainly by numerical and asymptotic techniques—between the following two phenomena: (i) the queueing behaviour at the second counter of the two-stage tandem queue and (ii) the queueing behaviour at a single-server queue with the same offered (Poisson) traffic as the first counter and the same service-time distribution as the second counter. This comparison makes it possible to assess the influence of the first counter on the queueing behaviour at the second counter. In particular we note that placing the first counter in front of the second counter in heavy traffic significantly reduces both the mean and variance of the total time spent in the second system.

Extremal properties of the FIFO discipline in queueing networks

1992 ◽  
Vol 29 (4) ◽  
pp. 967-978 ◽  
Author(s):  
Rhonda Righter ◽  
J. George Shanthikumar
Keyword(s):  
Likelihood Ratio ◽  
Service Time ◽  
Queueing Networks ◽  
Service Rate ◽  
Service Discipline ◽  
Single Server ◽  
Single Server Queues ◽  
First Results ◽  
Service Times ◽  
Number Of Customers

We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons:(i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means.(ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means.We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR.Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.

On a tandem queueing model with identical service times at both counters, II

1979 ◽  
Vol 11 (03) ◽  
pp. 644-659 ◽  
Author(s):  
O. J. Boxma
Keyword(s):  
Heavy Traffic ◽  
Queueing Systems ◽  
Single Server ◽  
Service Time Distribution ◽  
In Series ◽  
Mean And Variance ◽  
Queues In Series ◽  
The Mean ◽  
Service Times ◽  
Practical Implications

This paper is devoted to the practical implications of the theoretical results obtained in Part I [1] for queueing systems consisting of two single-server queues in series in which the service times of an arbitrary customer at both queues are identical. For this purpose some tables and graphs are included. A comparison is made—mainly by numerical and asymptotic techniques—between the following two phenomena: (i) the queueing behaviour at the second counter of the two-stage tandem queue and (ii) the queueing behaviour at a single-server queue with the same offered (Poisson) traffic as the first counter and the same service-time distribution as the second counter. This comparison makes it possible to assess the influence of the first counter on the queueing behaviour at the second counter. In particular we note that placing the first counter in front of the second counter in heavy traffic significantly reduces both the mean and variance of the total time spent in the second system.

Stochastic bounds for heterogeneous-server queues with Erlang service times

1974 ◽  
Vol 11 (04) ◽  
pp. 785-796 ◽  
Author(s):  
Oliver S. Yu
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Single Server ◽  
Shape Parameters ◽  
Server Queue ◽  
Service Times ◽  
Heavy Traffic Approximations ◽  
Multi Server

This paper establishes stochastic bounds for the phasal departure times of a heterogeneous-server queue with a recurrent input and Erlang service times. The multi-server queue is bounded by a simple GI/E/1 queue. When the shape parameters of the Erlang service-time distributions of different servers are the same, these relations yield two-sided bounds for customer waiting times and the queue length, which can in turn be used with known results for single-server queues to obtain characterizations of steady-state distributions and heavy-traffic approximations.

scholarly journals The stationary local sojourn time in single server tandem queues with renewal input

1999 ◽  
Vol 12 (4) ◽  
pp. 417-428
Author(s):  
Pierre Le Gall
Keyword(s):  
Sojourn Time ◽  
Stationary Regime ◽  
General Distribution ◽  
Single Server ◽  
Tandem Queues ◽  
Tandem Queue ◽  
Service Times

We start from an earlier paper evaluating the overall sojourn time to derive the local sojourn time in stationary regime, in a single server tandem queue of (m+1) stages with renewal input. The successive service times of a customer may or may not be mutually dependent, and are governed by a general distribution which may be different at each sage.

scholarly journals Multiserver queueing networks and the tandem queue model

1998 ◽  
Vol 11 (3) ◽  
pp. 377-390
Author(s):  
Pierre Le Gall
Keyword(s):  
Queueing Networks ◽  
The Other ◽  
Single Server ◽  
Tandem Queue ◽  
Case Scenario ◽  
Worst Case ◽  
Worst Case Scenario ◽  
Other Hand ◽  
Queueing Delay ◽  
Queue Model

Using a tandem queue model we evaluate the local “endogenous” (= internal) queueing delay in single server and multiserver queueing networks. The new concept of the apparent overall upstream queueing delay(as perceived by the downstream network) allows us to analyze the distribution of this local queue by interpolating between the distributions of the tandem queue (generated by a concentration tree) and the isolated G/G/squeue. The interpolation coefficients depend on the proportion of “premature departures”, typically interfering in the upstream stage and leaving the considered path without being offered to the considered local queue. On the other hand, local “exogenous” arrivals (from outside the network) require the introduction of the “interference delay” concept. Finally, in the case of single server queueing networks, we stress the need to extend the capacities of the buffers, by considering the “worst case” scenario and by using an “equivalent tandem queue” model.

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