The number of departures from a semi-Markov queue

1974 ◽  
Vol 11 (4) ◽  
pp. 825-828 ◽  
Author(s):  
D. C. McNickle
Keyword(s):  
Busy Period ◽  
Single Server ◽  
Single Server Queue ◽  
Server Queue ◽  
Customer Type ◽  
Interarrival Times ◽  
Busy Cycle

In this note an identity due to Arjas (1972) is used to express the distribution of the number of departures from a single server queue in which both service and interarrival times may depend on customer type in terms of the busy period and busy cycle processes.

The number of departures from a semi-Markov queue

1974 ◽  
Vol 11 (04) ◽  
pp. 825-828 ◽  
Author(s):  
D. C. McNickle
Keyword(s):  
Busy Period ◽  
Single Server ◽  
Single Server Queue ◽  
Server Queue ◽  
Customer Type ◽  
Interarrival Times ◽  
Busy Cycle

In this note an identity due to Arjas (1972) is used to express the distribution of the number of departures from a single server queue in which both service and interarrival times may depend on customer type in terms of the busy period and busy cycle processes.

On the integral of the workload process of the single server queue

2003 ◽  
Vol 40 (01) ◽  
pp. 200-225 ◽  
Author(s):  
A. A. Borovkov ◽  
O. J. Boxma ◽  
Z. Palmowski
Keyword(s):  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Single Server ◽  
Service Time Distribution ◽  
Whittaker Functions ◽  
Single Server Queue ◽  
Stieltjes Transform ◽  
Server Queue ◽  
Sequential Approximation

This paper is devoted to a study of the integral of the workload process of the single server queue, in particular during one busy period. Firstly, we find asymptotics of the area 𝒜 swept under the workload process W(t) during the busy period when the service time distribution has a regularly varying tail. We also investigate the case of a light-tailed service time distribution. Secondly, we consider the problem of obtaining an explicit expression for the distribution of 𝒜. In the general GI/G/1 case, we use a sequential approximation to find the Laplace—Stieltjes transform of 𝒜. In the M/M/1 case, this transform is obtained explicitly in terms of Whittaker functions. Thirdly, we consider moments of 𝒜 in the GI/G/1 queue. Finally, we show asymptotic normality of .

The total waiting time in a busy period of a stable single-server queue, II

1969 ◽  
Vol 6 (3) ◽  
pp. 565-572 ◽  
Author(s):  
D. J. Daley ◽  
D. R. Jacobs
Keyword(s):  
Waiting Time ◽  
Busy Period ◽  
Bessel Functions ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Common Distribution ◽  
Poisson Arrival ◽  
Server Queue ◽  
Common Distribution Function

This paper is a continuation of Daley (1969), referred to as (I), whose notation and numbering is continued here. We shall indicate various approaches to the study of the total waiting time in a busy period2 of a stable single-server queue with a Poisson arrival process at rate λ, and service times independently distributed with common distribution function (d.f.) B(·). Let X'i denote3 the total waiting time in a busy period which starts at an epoch when there are i (≧ 1) customers in the system (to be precise, the service of one customer is just starting and the remaining i − 1 customers are waiting for service). We shall find the first two moments of X'i, prove its asymptotic normality for i → ∞ when B(·) has finite second moment, and exhibit the Laplace-Stieltjes transform of X'i in M/M/1 as the ratio of two Bessel functions.

scholarly journals The transient behaviour of the queueing system Gi/M/1

1963 ◽  
Vol 3 (2) ◽  
pp. 249-256 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Distribution Function ◽  
Explicit Expression ◽  
Queueing System ◽  
Single Server ◽  
Single Server Queue ◽  
Transient Behaviour ◽  
Special Cases ◽  
Server Queue ◽  
Interarrival Times ◽  
Lagrange Series

SUMMARYWe consider a single server queue for which the interarrival times are identically and independently distributed with distribution function A(x) and whose service times are distributed independently of each other and of the interarrival times with distribution function B(x) = 1 − e−x, x ≧ 0. We suppose that the system starts from emptiness and use the results of P. D. Finch [2] to derive an explicit expression for qnj, the probability that the (n + 1)th arrival finds more than j customers in the system. The special cases M/M/1 and D/M/1 are considerend and it is shown in the general case that qnj is a partial sum of the usual Lagrange series for the limiting probability .

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